That’s exactly what I thought of. I always notice that when I see/hear/read British news reports, etc.
I knew an Irishman who once went into a lengthy diatribe about Americans using “the” in “in the hospital.” He was wound up kind of tight. That’s the same Irishman who went ballistic whenever we told him “Top o’ the mornin’ to you.” He’d angrily lecture that Irishmen never say this, but we’d say it to him any chance we could. 
Yep - tends to apply to institutions:
‘At school’ (and ‘at college’, ‘at university’), ‘In hospital’, ‘In prison’ = a student, a patient, an inmate.
‘At the ~’ just denotes being physically present at the named location - could be a visitor, a member of staff, etc (or a student/patient/inmate, but with emphasis on location, not action).
BTW: I keep seeing this thread title and wanting to report it as one of those ‘watch TV online’ spam threads.
This is also true of American English though we’d be more apt to say “in college” rather than “at college” to mean currently enrolled rather than physically there. “At the college” would mean physically there. However, I don’t think I’ve ever heard an American utter “at university” or “in university.”