I think it's probably because mascots are more prevalent in college sports. The Packers don't even have a mascot (as far as I know). Bango has been hitting headlines recently but I didn't know his name until this year. Bernie is hardly a mascot (at least in my terms of mascot). The sausages are my Brewers mascots. But Bucky Badger? Everyone knows Bucky Badger.
Plus I don't think people are actually referring to the team specifically when they say that, they're referring to Bucky himself. It's as if Bucky were running the entire Badger sports system and was every player on that field. As a student you have a little of that mascot inside of you, moving in and out slowly sometimes and aggressively others, until finally there's an eru...whoops.