Mario and Bowser are sworn enemies, plain and simple.
X and Zero started as Mentor and Disciple, then moved on to being friends or allies.
Samus and Ridley: Enemies (would be awkward but interesting to see interspecies rivalry though).
Link and Ganondorf: Enemies.
Alucard and Dracula: Dracula does not see his son as his equal counterpart. Same goes for Alucard. Enemies.
Cloud and Sephiroth: Mentor-Disciple complex. They become sworn enemies.
Squall and Seifer: In my little experience I had on FFVIII: I'd say this is the closest to the definition of rivalry. But falls to the overall dominance of Trainers Red and Blue
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You see, rivals are two persons who compete against each other for the same objective. Although their feelings can be strong, they don't necessarily feel the need to kill the other, but instead, to prove one's superiority over the other.
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Battle starts here.