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it's a common trend in campuses to see students take someone as a 'school daughter, school brother' etc. etc.
Does it suggest a secret wish for something deeper than the mere façade of 'just friends' as they always claim?

School mothers and school fathers are just "seniors" in secondary school( especially in the boarding house) who make it a duty to take care of a certain junior student.

The relationship is just friendship and most times it's just between members of the same sex.
It's in rare cases you have a male choosing to be a school father to a female student and vice versa.

Even at that it's still basically a platonic relationship.

But just like every other facet of life, some can have ulterior motives or secret agenda's as to why they choose to be a school mother or father to a certain person, which may include a sexual relationship.
But it all depends on the character of the individual.