It's interesting they don't have anything akin to the Greek system, or to those Ivy League secret societies, in UK universities, given how class conscious UK culture has historically been.
The system evolved here in the 1800s here. I wonder if it is actually an attempt to introduce a form of class or aristocracy to the US. The US is sort of tradition starved, since we're a much younger country. Maybe some US college students fanatically pledge allegiance to their Greek houses for the same reason many Americans get fanatical about our flag. We don't have the same kind of shared social institutions and history that exist in the UK. So the traditions that have developed tend to be clung to rather fiercely by some people here.
Or maybe it's simply because US universities are organized rather differently from British universities? As I understand it, the older universities in the UK (at least) are broken into multiple small colleges, and one is as much an alum of, say, Merton college as they are of Oxford. My understanding is students at those universities attend classes with and live in dorms with other members of their college, as a rule.
In the US, most universities are broken into much larger colleges that are focused on academics. For instance, at my university, we had a college of letters and science, agriculture, engineering, medicine, veterinary medicine, law, business and so on. There are dorms (mostly for 1st-year students), but you live with people from all the different undergrad colleges, and classes tend to be huge (at smaller, private colleges not so much).
In the US, you think of yourself as a graduate of the university first and foremost (there are also stand-alone 4-year colleges in the US, most of which are undergraduate only or focused more on undergraduate studies) and two-year community colleges that focus on transfer, vocational training, and community service. Of course there are the diploma mill online schools and "career" colleges now too, which exist to make money for the most part.
So maybe the Greek system was (or has become) a way for students to have a smaller community within a larger institution. The Greek system still tends of have issues with segregation by race and socioeconomic status, however. It's rather expensive to be in a traditional fraternity or sorority, and while members at a chapter will have their own biases in terms of selecting from applicants, preference is still given to people (called legacies) who have family members or strong references from alumni of the same house.