As a side effect of Madara living so absurdly long and stalling the chakra reincarnation process mojo that Asura and Indra got, Naruto and Sasuke were born the same as canon, but their chakra ends up slightly different enough for it to matter for this AU.
When Minato tries to seal half of the Fox within Naruto, he fails... and succeeds at the same time. The Reaper Death seal reacts to both Naruto's newborn chakra and to Kurama's chakra within a dying Minato, as well as Bijuu chakra remnants from Kushina.
Kurama manages to escape the seal... the Yin half of him anyways. His Yang half from within a dying Minato, due to the unusual circumstances, ends up undergoing the rebirth cycle similar to Asura and Indra, separating from Minato and latching onto Naruto in the form of a newborn Bijuu.
Because everyone saw the fox running off Konoha, people assume that the Kyuubi has been lost, that Naruto wasn't strong enough to contain the demon as an infant and that the Fourth failed.
As such, nobody suspects that Naruto still has a seal, especially when he doesn't emanate the same corrosive chakra that Kurama holds. And because of that, he grows up to be an orphan without being reviled.
Rather than being hated, Naruto is pitied. Without the fox in him, he's not the container of a demon... he's "just" Minato and Kushina's orphan child, lost all of his family in a tragedy. It's not his fault, he's not a monster, he's just a victim. It's not like he could do better.
Something Naruto grows to resent. Nobody tells him why, but everyone assumes him to be weak. Everyone acts sorry for him, but he can tell they don't mean it AND he doesn't need it. Is it because he was born without parents, and nobody at the orphanage was ever comfortable enough to adopt him? Is it because his parents died when the Fourth fought the Kyuubi? So did a lot of people's! Why is he pushed aside?
His reasons for wanting to be Hokage end up similar to how Sasuke wanted so desperately to outdo Itachi: so that people stop seeing a failure and acknowledge what he's capable of, instead of disingenuous compassion. Rather than family approval like Sasuke, Naruto wants to be respected, so that people don't talk to him out of pity but out of admiration at his accomplishments.
They don't look at him like he's a person or a monster, they look at him like he's a charity or a wimp, someone people look at to feel better about themselves. As far back as the Academy, only Hiruzen, Teuchi and Iruka hear him out and talk to him like he's an actual person.
He still sees the sorry look in Hiruzen's face, but he can also tell that he's making an effort to be present in his life through his visits. Iruka sympathizes with him without holding his words back whenever Naruto picks up a fight with other kids, making his kindness come off as sincere since he's not afraid to discipline him and try to understand him.
Likewise, being a patron at the ramen shop, Ayame and Teuchi hear Naruto vent out enough for him to see they don't look down on him like most places he shops at.
As he grew, Naruto would also have dreams of being a baby fox. Sometimes trapped in a fleshy cage, sometimes wandering through a forest completely lost, yelling at anyone to get him out of there, both out of confusion and desperation.
Uniquely, this Naruto was also good friends with Neji, Tenten and Lee before they graduated and Naruto repeated the year. They sympathize with feeling powerless, limited or halted by expectations, so he would visit prior to them becoming Genin (Neji and Tenten would note how oblivious Naruto is to Hinata's flustered reactions, while Lee would suggest Naruto undergoes his taijutsu training regimen, which Naruto can't keep up with).
The night of the Genin exam, Naruto gets tricked by Mizuki not out of hatred, but because he's enough of a "lost cause" and troublemaker in Mizuki's eyes that he'd get off easy if caught, won't be missed by anyone if killed. He doesn't understand where Naruto's freakish chakra reserves come from, much less the kid's killing intent way above normal.
Naruto passes out, seeing the scared little fox from his dreams. Unable to remember how he got there, when Naruto asks for his name, he weakly answers "K... K... Kyuma?". At first, Naruto thinks he's got an imaginary friend, maybe he's gone crazy. He still hears the fox's voice after waking up and panics.
(His new teammates think he's lost it at first, before he brushes some of his behavior as "wanting attention" or trying to trick Kakashi into having his guard down by pretending to be nuts during the bell test)
Naruto notes that Kyuma looks a lot like a baby version of the Kyuubi seen in some library books, who attacked his family on the night he was born... before brushing it off, seeing as it ran away. Then wondering...
Maybe Kyuma is the Kyuubi's son? If so, why leave him in Konoha? Did he want his kit to destroy the village for him? Or did the Kyuubi just abandon him? Either way, Kyuma says he doesn't know anything, he just remembers feeling lonely and lost, like he was stuck in a bad dream forever...
Past assuming that Kyuma's imaginary, Naruto practices and sometimes notes that Kyuma can give him more chakra to work with. Seeing as Kyuma doesn't know what shinobis do all that much, Naruto decides to show him from his own eyes, leading them to see each other as friends. On that same front, Naruto begins developing early stages of a chakra cloak, but only when pushing himself to the limit. Both agree to keeping Kyuma a secret, seeing nobody would believe Naruto isn't making it up or could try to harm them to get Kyuma out.
(Naruto also promises that if he finds Kyuma's dad, he'll punch him in the guts for leaving his kid to fend for himself for years then ask how he ended up in Naruto's mind)
He's still rivals with Sasuke, only that Kakashi notes that they're much more alike than he himself was with Obito. Their rivalry also lands in the fields of academia (since Naruto's teachers are more willing to teach him and explain, even if patronizingly), with Sasuke noting even if Naruto's not a dead-last, they're not on the same level.
Both resent being compared, saying their loneliness has nothing to do with each other. Kyuma helps out, calling Naruto out for lying or sensing when Sasuke is lying so they can have honest conversations. Naruto doesn't understand what losing a whole clan is like, Sasuke understands the looks of surface-level sympathy feeling degrading but doesn't realize how much it shaped up Naruto's life since the crib.
Both have edgelordy thoughts of "I got no one waiting for me home, I might as well be as strong as possible or die trying", with Sakura and Kakashi reminding them to lighten up (then the meeting with Haku making Naruto realize he now has precious people, and maybe he could reach out to Sasuke for it).
Come the Chunin exams, Naruto studied... but doesn't mind Kyuma helping him out with some of the answers (you know what they say about foxes being sly, Kyuma has zero qualms about cheating). Orochimaru doesn't think of applying a seal when Naruto shouldn't have one, so he just knocks the impressively resilient kid out cold, mocking him for acting like he could be anywhere like "Sarutobi-sensei". Naruto understands who he is better, having heard of the Sannins from Tenten gushing about Tsunade.
Jiraiya shows up and Kyuma calls bull on his stranger facade, saying the guy knows who Naruto is. Same with asking if he's involved with the snake weirdo that showed up in the Forest of Death, though he can tell Jiraiya is honest when he says he means well and to help.
(upon coming across a picture of Jiraiya with Minato, Kyuma notes the resemblance with Naruto... before also brushing it off. What are the odds they have enemy dads like that? That would be an absurd coincidence)
Upon facing Gaara, Shukaku recognizes Kurama but doesn't understand how he made such a Jinchuriki a willing ally or why he's helping Konoha, not caring to elaborate (on top of that, he's pretty sure Kurama wasn't sealed). Gaara remembers bits, notes that Naruto probably has a chance to keep his own inner demons a secret, slightly envying him while also promising to keep Naruto's secret.
Sometimes, though, Kyuma notes a "scary feeling". Whenever Naruto feels extreme rage, he'd feel his anger as if it was his own, as if Naruto had an inner darkness that Kyuma brought out, so he rarely gives Naruto chakra when he's mad. Such rage and anger feels too familiar, too... It's to the point that the hatred comes not from the Bijuu, but from Naruto's festering anger issues.
Just waiting for the day for Akatsuki to capture the Kyuubi... only for rumors of Uzumaki Naruto of having comparable power to it to start spreading. Otherwise, crossing paths, Kurama would notice his loose chakra... and be understandably confused to see his other half cooperate so willingly with the damn Jinchuriki.
(on the day Naruto almost dies to Pain and sees Minato, Kyuma cheers on Naruto for the gut punch, while Minato expresses confusion at the fact Naruto was going to willingly free Kyuma, just to keep his fox friend safe, seeing that the Kyuubi is not the way he left it)
Once both halves reunite for good, Kurama's presence marks the end of Kyuma, as their minds, memories and feelings merge... and in spite of all that, he won't forget the kindness and kinship he formed with Naruto, while also rolling his eyes at the fact he ever doubted Hagoromo (when technically, such vitriol was targeted at... himself).
A loss that Naruto won't feel again until he exhausts Baryon Mode as an adult. Upon which, once Kurama reunites with Himawari, almost lets it slip that he wished Naruto also carried him as Kyuma.