I find it absurd that a lot of people have voted "yes." The answer is assuredly no, particularly if one can't agree on who that 2nd rival is...because you can't have 5. Look, conference games are conference games, and you grow to dislike pretty much everyone you play every year. But rivalry?
I'd say the only other team that could come close historically is Illinois, but the Michigan thing got out of hand maybe 80 years ago? So, yeah we still do the turtle thing, but Michigan is too much an arch-rivalry that all others simply don't measure up. We compete with Michigan on every single level of our pointless lives, from football games and recruits to whose city sucks less Cleveland or Detroit.
Wiscy and MSU are competitive teams and those have been entertaining matchups over the past half decade - 15 years or so. But go back to the 80s...Iowa was a big matchup every year...in the 70s it was only Michigan...in the 60s Michigan St. was a big game...beyond that you're getting into the area when people don't remember.
So, no. No 2nd rival. Penn State is our doppleganger that we are trying to avoid. Eventually we may acknowledge them as a 2nd rival, but I can't say when that will be.