Which schools in United States offer a similar experience?
Everyone gets sorted into a House at Harvard, but it doesn’t happen until sophomore year. Vanderbilt also does it this way. Like Yale, Princeton sorts first-years into colleges before they arrive on campus, but Princeton’s version differs in that many juniors and seniors don’t live in the residential colleges, and because of the eating club system there.
That’s somewhat similar in some ways but which schools other than Yale, offer an established residential college system like oxford and Cambridge, where you live and eat and play and often take many classes at your own college and have an intimate family like small community for all four years.
Unlike Oxbridge, despite what Wikipedia says, courses are not specific to the residential college at Yale (or Harvard or Princeton). Some very small number of courses are held in the residential college as a supplement, not in lieu of.
Rice is another.
Franklin and Marshall does something like this, as does Union College, though they don’t actually have classes in their houses. These colleges are small LACs, so there is no need to have dedicated classes for one housing community. There are probably other LACs that have similar arrangements. You may want to consider a wide range of colleges, not just ultra-selective ones.
In terms of looks, Sewanee often gets compared to Hogwarts. We visited; beautiful, beautiful place, but it’s not right for D19 (very small). The honors kids wear black robes, a la Oxbridge/Hogwarts.
Purdue’s honors college sorts into houses and has a house cup (complete with an app that tracks points) ; )
SMU
Rice does this.
UC San Diego has a bit of this… the large university is broken down into 6 colleges, each with a housing area on campus and its own set of breadth requirements. You can major in anything from any of the 6 colleges, although some fits are better than others. Kids rank their choices when they apply to the University and are given their assignment when they get accepted to UCSD. Not sure if you can swap once you are in. From what I understand, upper classmen typically don’t live on campus. So you’d have your breadth requirement courses mostly within your college but other classes could be with anyone. (This was my oldest kid’s ‘also-ran’ school so we looked into it heavily but she didn’t end up going, so I could have some details wrong).
UC Santa Cruz is divided into individual colleges, each with its own mini program, personality, and specialties --I’m not sure how they “sort” however. I doubt they use a sorting hat, though.
Durham would qualify but it’s in the UK 
Rice is the one that seems closest to the sorting hat situation.
Franklin and Marshall
Rice indeed comes closest to Yale/Oxbridge residential college system.
And Rice even has a pretty unique exchange program with Cambridge (https://ccl.rice.edu/students/fellowships/rice-nominated-fellowships/cd-broad-fellowship/)
I lived, ate, played, and took classes in the Residential College at the University of Michigan. In my experience, Michigan was a small LAC.
LOL! In my mirror, i’m a genetic wonder but that doesn’t effect my genetic make up.
Wow, “a small LAC”, with 30000 of your classmates, just don’t see it. You really need to actually attend a small LAC to understand what that is like.
I think that @ChoatieMom’s point is that a residential college system at a large university can create a smaller, more intimate community… which is also part of what the OP is seeking.
Caltech also has a House system. https://en.wikipedia.org/wiki/House_System_at_the_California_Institute_of_Technology
Only one of the houses has faculty who live in the house, in case that feature of the Yale-style residential college is what you are looking for. Also, Caltech is a lot smaller than others that have been named.
Only rarely are classes located in the Houses, and then they are available to all students.