By the meaning you are using for “exclusive,” that wouldn’t apply to Boston College, Georgetown, or Notre Dame, referenced in the original post using the term. They do not have moral standards or expectations for students.
Maybe that post just meant “academically selective.”
I was thinking more of campus life, segregated dorms, dress code, students must refer to each other as Mr and Ms.in class, dry campus, no sports, academic expulsion for a week of unexcused absences in any one course. It appeals to a small segment and excludes many, because they don’t want to abide by such discipline.
That’s not to knock those who don’t go, I don’t really want my daughters to go there. Although not due to most rules.