What people fail to realize is that Wisconsin fans often wear black hoodies with Wisconsin in Red/White lettering. They also wear a navy color similar hoodie. It is not that Wisconsin just "wears red and white". They really have no unified dress code like you find at Nebraska. Wearing red would have been just as fine.
Also, we can argue who the "original" big red is until the cows come home, but only one school in the nation that I know of has the name "The Big Red" (albeit unofficially). This school is neither Nebraska or Wisconsin. Hell the local high-school there is even called the Little Red.