UW has 2.5 rivals
Minnesota
Iowa
Sort of Nebraska (only been played 16 times)
NW isn't really a rival. It's a team that every loss to them makes me want to put a fist through a wall because the game is always stupid. I don't get the sense NW specifically cares, and I don't feel like UW thinks of the game as anything but a normalcy in victory and an affront if there's defeat.
Michigan and OSU are weird cases. I think at points the games are competitive for some rivalry-esque fire, but those are isolated. For a time, UW was doing decent against OSU, couldn't touch Michigan. Now OSU is OSU, and UW-Michigan is just a grab bag with each team's volatility.
MSU was a thing for a time, but if you've played three times since 2012, it's hard to keep that fire going, even if that game last year made me want to burn Spartan Stadium to the ground.