He didn't exaggerate anything. He assumed that because someone doesn't like Jazz, it must be the wrong kind of Jazz. That's almost a perfect fit for the fallacy, and that's what made it funny. If you don't like the joke, that's okay. Doesn't mean it's not a joke.

I edited to put the joke part back, because without it your comment's part about joke wouldn't make sense, and that's bad reddiquette.