Rivalies in sports are fun and look forward to be their fans. Making one of the leagues biggest team your primary rival benefits the Tigers greatly, there is no rule that says a team like the Yankees can't have more then one rivaly going at once so I see no problem with the Tigers making the Yankees their primary rival.
yes, the tigers need to look at the yankees as their main rival. Just because the yankees do not see them as the main rivals back, does not need that the tigers do not have to, they can still be fired up about playing the yankees and winning against them.
Rivalries have to work both ways, almost by definition. Rivalries are almost always between teams or organizations that are comparable in skill and reputation, which would make sense in this case, but both teams (and their fans) are far more important in forming the opinions and rivalries that last for generations.
The Detroit Tigers should not consider the New York Yankees to be their primary rival, even though the Yankees already have a rousing rivalry with the Boston Red Sox. The Detroit Tigers should consider the Red Sox to be their Primary rival because the Red Sox have been dominating the American League in recent years.
I guess I can see how the Yankees are in this conversation, but Boston did win it all last year. Plus, Boston were the ones to knock out Detroit in the playoffs last year. I think that would make Boston more Detroit's rival than the Yankees, who did not even make the playoffs last year.