I have a question that's on a related note. Does each school only have one training location? For example, is Kaer Morhen the only place Witchers of the school of the Wolf were trained or were there other locations? I've never read the books and I can't remember it ever being explained in any of the games and the Wiki is massively out of date and in dire need of some attention. If there were other training locations for each school its entirely possible there are far more Witchers than we know about.
That was my impression.
I get the impression each castle is more or less like a Dojo.
So Geralt being of the Wolf School means he's from Kaer Morhen.