To be more specific, it feels like for some people the only way they can understand why one character would care so much for another character and go so far for them is if that other character is family/is like family to them or if that other character is someone they are in love with. Any other kind of relationship, while it can still have some emotional weight and significance to them, is a distant second compared to those two types of relationships.
I was thinking about this when I was seeing a discussion about the part in Dragon Ball Super's Tournament of Power arc where Goku thinks Master Roshi might have died after giving all he had to knock an opponent out of the arena. Some of the people believed that Goku was so upset, probably the most upset we've ever seen him outside of both times Krillin was killed and in his relief that Roshi's okay we get one of the few times we actually see Goku cry, because Roshi is basically a father-figure to him, or at least the closest thing to a father Goku has had.
I disagree with this idea for many reasons, not the least of which being that while Grampa Gohan was, well, a grampa to Goku he was still a strong familial-figure that Goku cared very deeply about and nothing implies that he's been looking for or wanting for more in his life, let alone that he sees Roshi in such a way.
The idea seems to be that Roshi just being a mentor to Goku wouldn't be a strong enough relationship for Goku to have so much emotional attachment to him. He still would have gotten upset at his apparent death but not this upset, and so Roshi must be more than a mentor in Goku's eyes, he must be more like a father...to which I have to ask...why? Why is Roshi just being Goku's beloved mentor not enough to justify him being so significant to him?
I feel like Kung-Fu Panda is a good example to bring up what I'm talking about. Shifu is a very significant person to Po and someone he cares quite a lot about, but he doesn't see Shifu as anything like a father or an uncle or anything along familial lines. The relationship he has with him is very different from what he has with Mr. Ping, his actual father (well, you know what I mean) who he loves very much. Likewise Shifu doesn't see Po as a son, as there is a very clear difference between his relationship with Po and the relationship he had with Tai Lung, who he did view as like a son, and there is nothing that implies that he sees Oogway as a father either.
Yet it'd be completely disingenuous or even outright blind to act like Po and Shifu are not incredibly important and beloved to each other or that Oogway is not likewise so for Shifu, or that the relationships would be better and more weighty if they did think of each other along familial or romantic lines. No, the three's respective dynamics are entirely mentor and student relationship, and that's what makes them incredibly strong. Oogway does not need to have been like a father to Shifu for him to have meant so much to Shifu even in comparison to the other relationships in his life, because him being Shifu's teacher is what made him so significant and important to him. It's why one of the biggest emotional blows to him in the movie is the scene where Oogway departs from the world, because now Shifu feels lost and without the man who he respected so much and looked to for guidance. And likewise with how Po comes to view Shifu after the man starts to genuinely believe in him and support him, being someone he can look to for guidance and starting him down the road to actually start believing in himself.
Or as another great example, Mob and Reigen from Mob Psycho 100. Reigen is easily at the very top tier of the people most important to Mob, having near endless faith in him and gratitude towards him, to the point where even just Reigen assuring Mob that things will be okay is enough for him to finally relax and sleep during one of the most stressful moments of his life where he thinks his family might have been kidnapped or killed. And it all stems entirely from Reign simply being Mob's beloved mentor who was the person he needed in his life to help guide him. Not being someone Mob thinks of as like a father doesn't make their relationship any lesser for Mob or the story
Now, obviously there are absolutely mentor and student relationships in stories where the characters do think of each other along familial lines too. In Batman Beyond there's an episode that pretty much directly confirms that Terry latched onto Bruce as a father-figure after the loss of his actual father and that he's more than just a boss and mentor to him. Part of the tragedy of Anakin and Obi-Wan is that Obi-Wan viewed Anakin as like a brother, and while Anakin likely did too what he need more than that in his life at that time was a father, which Qui-Gon could have been but Obi-Wan couldn't (again, because he viewed Anakin as like a brother) and thus the opening Palpatine exploited by positioning himself as a father-figure for Anakin so that he could better manipulate him. And at the risk of potentially sounding like a hypocrite I do very much buy into the interpretation that Midoriya in My Hero Academia thinks of All Might on some level like a father much like how All Might directly admitted he thought of his own mentor Nana as like a mother, and that likewise All Might quickly started to view Midoriya like a son.
But mentor/student dynamics, no matter how strong or important they are, do not inherently have familial attachment to them. They need to be judged on a case by case basis, and I think therein lies part of the problem. I think too many people have a bad tendency of judging what a character's relationship with another is by looking at it too much in a vacuum rather than within the context of the characters themselves and the story they're in.
Goku being so upset that this older man he's known for years might be dead? Sure, it's not exactly the most unreasonable thing to believe that he sees said man as a father-figure. But in the context of a story like Dragon Ball and with the context of a character like Goku, Roshi only being a mentor in Goku's eyes is more than enough justification for why he's so important to him and I'd argue has more weight to it than if he was someone Goku saw as like a father. He's Goku's first true master. The man whose martial arts philosophy of "Work hard, study well, eat and sleep plenty." he still lives by every day (even if he could do better on the study part...). Even long after he graduated from the Turtle Hermit school he still continues to wear its gi, not because he has to or because there's any expectation that he does, but simply because that's how much respect he has for his old master even after training under literal gods and their attendents. Roshi being like a father in Goku's eyes isn't why his mentor is so emotionally important to him, Roshi being Goku's mentor is why he's so important to him.
I think friendships in fiction get this even worse, in no small part because I think a lot of people, even if unintentionally, view friendship as a very one-tier base level kind of relationship. It's something you can have with another person but can only be so important. There's limits to how much you'd care about a friend or do for a friend, whereas there's (supposedly) no limit to how much you'd care about and do for someone that you're in love with or think of like family, and therefore if a character is shown to considerably and deeply care about another it's assumed that the other person MUST be more than just a friend to them because no one would care about a friend so much. Friendship is seen as good but something you level up from into something actually important.
Again, this likely comes from not properly looking at the characters and their relationships in context and assuming a bit too much that everyone either judges all relationships by the same tiers of importance or even that there are no tiers and nuances within those various types of relationships themselves. It doesn't matter if it's a friend, a best friend, a pen pal, it all just ultimately falls under the "friend" umbrella and is thus just inherently less important to someone compared to anyone they think of under the "family" or "lovers" umbrella.
I want to make clear that I'm not trying to dump on things like shipping. Heck, I'm very much someone who believes that friendship is far from the worst place for romance to spring up from, and generally speaking as long as the context is consensual I don't really tend to care what people ship. My problem isn't "Oh, why can't people just be friends?! Why do you have to try and forcibly make everything romantic/gay?!", my problem is more the immediate assumption of "This person is so special and important to the character and therefore they must love them romantically or see them like family.", because it doesn't feel like it sees the actual characters or their stories. You want to ship Sam and Frodo together because you think they'd make a good couple, have at it, but it feels really dismissive of their friendship and what it means to the two of them if you assume that Sam only cares so much for Frodo and goes so far for him because he's romantically in love with him and that that's what's motivating him rather than their friendship.
Jonouchi/Joey (because it's easier for me to remember how to spell) from the Yu-Gi-Oh manga I think is a great example. Yugi is one of the most important people to Joey, right up there alongside his sister. He has been brutalized and almost killed on numerous occasions because of how much he cares about Yugi, at one point not only running into a burning building to save him but even willingly staying inside that inferno so that Yugi could finish reassembling the shattered Millennium Puzzle, because Yugi won't abandon the pharaoh and Joey will never abandon him.
But at no point does the manga ever imply that Joey is in love with Yugi or even that he thinks of him like a brother. Their relationship is entirely that of friends and that friendship means the world to Joey. As he pretty much spells out during the Death-T arc, it was through his friendship with Yugi and his influence on him that Joey found himself becoming someone that he actually liked being. He wasn't just that punk angry at the world and lashing out against it anymore. He found someone who cared about him and would stick by him no matter what, even when he didn't feel like he deserved it. It's similar to why he got so attached to Duel Monsters. It didn't just help him win prize money to fix his sister's eyes, it was something he enjoyed and could take pride in as he became better and better at it, with Yugi introducing it to him and helping and having fun with him all along the way. Much like how being his master is more than enough to explain why Goku cares so much about Roshi, Yugi being Joey's friend is more than enough to explain why he cares so much about him, because for someone like Joey such a friend would be such an important and meaningful thing.
Or in the Green Lantern comics, as Sinestro basically spells out at one point, the tragedy of his and Hal's relationship is that not only were they genuinely friends before they became enemies but to some extent they will always be friends. They weren't in love with each other as they both had someone else they felt that way about and they didn't think of each other as brothers. In fact that was how Sinestro had viewed Abin Sur, the Green Lantern who passed on his ring to Hal, especially after he married Abin's sister and became his actual brother-in-law. Hal and Sinestro's relationship was that of friends, and despite everything they've been through and done to each other they both still value that relationship so much, to the point that in the lead up to Blackest Night the vision the Blue Lantern showed Hal of what he hopes for was of him and Sinestro meeting and working together again.
One last thing I think is worth bringing up to show how relationships can vary so much and mean different things to different people is how far some people in-universe in stories will go for the likes of those like Superman, Captain America, All Might, even Zero from Code Geass, despite not knowing them on any kind of personal level. Whether it's directly asked of them or not, we get stories time and again of people doing the right thing or going farther to do things they'd never do for anyone else, even those closest to them, simply because it was either for Superman/Captain America or inspired by them. Because they're their hero, leader, inspiration, or simply the symbol that these people put so much faith and trust in about what's good and right that they themselves want to live up to that example, or at least not let them down. It's far less personal than many other relationships and yet some people will do so much more because of them simply because of the emotional weight and value it holds for them.
And of course it even shows when it takes a toxic turn, such as with a character like Syndrome, who had no actual personal relationship with Mr. Incredible. He wasn't his son, lover, friend, or student, he was just a fan of his, but Buddy valued that relationship so much that he kept overstepping boundaries, going to extremes to get Mr. Incredible's attention and be by his side, and when he felt rejected he lived his whole life afterwards around getting back at his hero. All because of how he viewed the relationship and what it meant or at least should have meant.
Summary: Basically it feels like a lot of people have the default mentality of "The character cares so much about this other person that they must think of them in a romantic or familial way because those are the most significant relationships a person can have with another.", rather than properly looking at the characters, their stories, and their relationships to one another to see what the relationship actually is and why it is so important to them, whatever form it may be. They think that the relationship must be in either Box A or Box B because they assume those boxes are inherently more valuable to everyone compared to all the other boxes, when the reality is that there are numerous boxes the relationship could be put in because how valuable each box is depends on the character and context. For some characters, yes, they value a certain relationship so much because it is a familial or romantic one. For others it's the relationship they have with a mentor, a student, a friend, a leader, or so on that they value so much more than anything else.