I'd still say Liverpool is the big game for United fans.
And yet Everton/City, there's basically never ever been anything, even when they were close together in the league. So it's not just Mancs hate Scousers and vice versa.
Arsenal vs United from the late 90s to the mid 00s was a big one that had nothing to do with geography.
If you're really old then you might view Wolves vs Stoke as a rivalry but I'm only a little bit old. I don't like them because of their mutant fans and frequently mutant football, but it makes no odds to me what they do.
Some Albion fans kid themselves that Villa are their "real" rivals, dream on. They don't care about you, lads.