The terms "university place", "college place", and especially "student place" are rarely used in the United States (for example in this Washington Post article), where there are no federal or state policies or plans regulating the total number of university students and where private and even state universities usually do not have a predetermined number of places available. This means that there is no such concept as a place waiting to be "filled". State universities often admit all applicants who fulfill certain criteria, and private universities take as many students as they are interested in, adding faculty if necessary. Universities and colleges sometimes talk about how many places they have available for certain programs or for certain kinds of students, but then they use the general term "place".
Instead of saying that a student has received a university place, as in the United Kingdom, it is usual to say in the United States that the student has been admitted, but the expression "she got a place at <university's name>" using the general term "place" is of course also used. Even more common in common parlance is to say that someone "got in" or "got into <university's name>". Less common is "get a spot".
Discussions in the United States about the total number of students in the entire country or an individual state use terms such as "number of college students" or "enrollment".