Fansites can give out these cups in contests, but I think they avoid that because they really don't fit to non-sport-related contests. In the past, these cups were more common to be obtained in football tournaments, but I don't know if any fansite still organizes them.
As for the Warrior Trophies, I believe it is a choice made by whoever is organizing the contest. At least at the time I was helping to organize contests for fansites, there were no such limitations.