I think that the question being asked here is more of the form:
Given a Sports Team Name of the form - 'Location Name' + 'X' (e.g. St. Louis Blues, L.A. Lakers, Miami Marlins), is it appropriate to refer the component of the name represented by 'X' as a 'nickname.'
I would be inclined to say no as a 'nickname' implies that it is a shorthand for a 'formal name.' (e.g. Jim instead of James.) The name of the NHL Hockey Team in St. Louis is not anything other than "The Blues," though nicknames for that exist. In fact, the only team I can think of more known by a nickname than their real name is the Knicks (Knickerbockers.)
If you're looking for a word or phrase to identify that specific part of the name, the only two options I can come up with are simply "team name" or "brand."