I am having trouble understanding how a non profit can even get a Trademark.
I don't think the school itself has the trademark. I could be wrong, but I think that it belongs to the Athletic Association which is possibly "for profit".
In college sports it can be hard to figure out who actually owns what. At the University of Georgia last year, there was a turf war between the University of Georgia Foundation and the school itself. Most of the university trustees were also board members of the foundation so it was hard to figure out exactly who ran the school or the athletic program.