I imagine the main issue is the schools being different places. Kaer Morhen should be the single "school building" of the witchers. Makes sense with Geralt being a White Wolf too - nickname makes more sense if not everybody around is a wolf.

But there are clearly differences between the people with different medalions - like the cats mostly becoming assassins or swords for hire. I'd be really interested in him explaining the different medalions someday.

Edit: also makes me wonder what schools/medalions the others had. Never thought about this one, not sure if it is mentioned.