I think you're off base by assuming that characters that ally with each other can't be rivals. I think that actually goes against the concept of being rivals. Rivals are generally competing for the same thing (or at least they're competing on very similar terms), but that doesn't mean they don't team up for a common goal. Phoenix and Edgeworth do this all the time but they're obviously rivals.
Mario and Bowser aren't rivals in like, the Mario platformers, but they definitely are most of the Mario RPGs, albeit to be fair it's a bit of a one-sided rivalry because Mario doesn't talk. It's pretty obvious that Bowser sees Mario as a rival and not as a mortal enemy in the RPGs, though.