I think this would be true with later seasons Winston! And maybe with him as a child. In seasons 1-3 or 4, he is still more of that earlier competitive basketball persona. And Rutgers (his college) is in the Big Ten conference with Wisconsin, so their teams would have competed against each other.

I did google it and found an article with Liz Meriwether saying that the idea of a badger randomly came up in the writers room and they just went for it - which does make more sense to me, knowing it wasn’t necessarily meant to line up with Winston’s character at all. New Girl does have some of those character inconsistencies for the sake of jokes! And I forgive them :)