I can see how one team can be seen as the rival of several teams. How many baseball teams think beating the Yankees is the best thing they could do and, after all, what other team would the Toronto Blue Jays see as a bitter rival, more bitter than the Yankees? But everyone knows that the Yankees true rival is the Red Sox. But the way that many teams with multiple "rivals" get that way is typically by winning getting the reputation of winning, so it's a big deal to beat the King of the Hill, or at least some other team you think is ordinarily better than your team. (Maybe a team's Fan Survey should have a big jolt upward after a "big win" against a superior opponent, and a similar big drop after losing to an inferior team. But that's for a different Suggestion thread...) Here, I'd hate to have to beat a perennial powerhouse to get credit for beating a rival. In the case of BB, it is certainly better that a rival also be a team that would play a close and exciting game, one that is similar in roster strength.
But your main point I think is generally a good one, and part of what makes the rivalries in the game often the point of criticism since they very often don't necessarily make the most sense.
Don't ask what sort of Chunks they are, you probably don't want to know. Blowing Chunks since Season 4!