= do ivy League schools have individual colleges the same way Oxford and Cambridge do? =

Not really in the same way

US Universities have different colleges by study area. Liberal arts, engineering, medicine, agriculture, etc

It isn’t the same as the way Oxford and Cambridge operate with several colleges under one umbrella university but not split into specialties

A lot of US universities do have a "residential college" system modeled on the Oxbridge college system. Rice is one example; it was deliberately set up to mirror the Oxbridge colleges. (Their top-level academic departments are called "schools
Also, sometimes a big-name school will absorb a smaller school and preserve the smaller school as a satellite campus, or a separate undergraduate academic college, often handling the liberal arts program

And sometimes you have the historical women's colleges like Radcliffe, which was gradually absorbed into Harvard, and Barnard, which is still a women-only affiliate of co-ed Columbia

Our colleges are an academic division. They'll usually have their own dean, budgets, etc. My university even had few colleges with higher fees. College of Engineering and Design, College of Business Administration, College of Human Ecology, College of Arts & Sciences and so on. Within those you have the different academic departments. The department or school of Architecture, the Department of English, the Social Work department, and so on. There are also often honor societies within these smaller departments which are open to students with a certain grade average. These and maybe some dorms take the place of the social aspects of your colleges

The names tend to vary (though "college") is the most common division

The big exception for all 3 ivies I've attended in various capacities is that there tends to be one or two "colleges" that are so big and diverse that they might as well be renamed "College of Not Engineering and Not Anything Else We're Famous For."
So for example my alma mater had the "College of Arts and Sciences", which in practice could have been renamed the "College of Not Medicine and Not Engineering" without changing anything but the letterhead

The Yale version is sort of Oxbridge light, which makes sense because of the way the systems came about. Other than the more modern ones, Oxbridge colleges started out independent, as I understand it, and eventually unified under the relevant University. They therefore have a lot more autonomy from the University, separate funding sources at least in some instances, etc. Your degree is granted by the college. Yale colleges provide undergrads with a cohort of sorts, and have some administrative and advisory staff, their own dining halls (that are open to all) etc. but they don’t