Yeah, I don’t get his angle on this. On the other hand, more than one athlete has used the same nickname before. “The Little Ball of Hate” is a fairly particular nickname that was created for Pat Verbeek and is currently used to refer to Brad Marchand. I don’t think either player tried to get a trademark on it. 
People in New England don’t use Tom Terrific wrt Brady very often. Unless his people have a huge campaign planned it seems like a waste of time and something that will just alienate part of his target audience. I don’t know how many Mets fans would ever cheer for him, but lots of casual fans might view this as a petty act.