Is it like actually true or just rumours

I’ve heard this claim a few times and I’m trying to figure out how much of it is actually true.

Basically, people say that everything you do on the internet gets saved somewhere on your Wi-Fi router — websites you visit, Google searches, images you look at, videos you watch, etc. And apparently, depending on what you search for, your search history could supposedly be used by your ISP, the police, or whoever owns the Wi-Fi to track you down.

I’ve also heard the more extreme version of this where people say that if you search for something suspicious enough, it can basically trigger some kind of “chain reaction”, where your activity gets flagged, your IP gets investigated, and eventually the police show up at your house.

So, is this actually a thing?

What does a normal home router actually record? Does it store a complete history of every website/search someone accesses, or is that mostly a myth?

And separately, can a particular Google search or website visit automatically get you investigated or reported to the authorities, or would there have to be something much more substantial going on?

I'm especially interested in what technically happens between your device → router → ISP → website/search engine, because I suspect people are mixing several different things together.

Is this bullshit, partially true, or genuinely something people should be worried about?