When you are referring to this "variant", are you referring to maximal matching, or bipartite matching when the number of people and schools don't equal?

Not the parent, but I think they refer to the fact that with admissions, matches are permanent (once a student accepts a school's offer, said offer can't be rescinded), while Gale-Shapely requires provisional pairings until all the rounds have been gone through and a final result is available.

The issue is that couples want to be placed together. That makes everything trickier than if people just placed independently.