Yes but they do anything to get audience and hearing that name, and they would milk for all it's worth in insulting a adult that go around calling himself Dr. Eggman is prime material for very insensitive jokes about his mental state.
I mean some participants are implied to be Neo-nazies, the game does not pull punchies the people that enjoy watching are just as evil and deprave as the participants while having a place call Asiantown with all the stereotypes about Asians in the planet on it.
So yes the narrators of the matches, Kessler and Howard (I think they name are) would insult Dr. Eggman in the worst ways possible because they are that horrible and the Eggman can't stand the fact of people laughing at his machines or him much less.