Mario & Sonic. The Olympic games series are NOT adventures. I think something like the Mario & Luigi games, with each character's actions in battles and the overworld mapped to a single button. Using mario's jump and Sonic's spindash for overworld puzzle solving, special attacks combining their abilities like having Mario hitting Sonic into enemies with his hammer like a golf ball, Sonic making a tornado and Mario shooting a fireball to turn it into a fire tornado. Bowser and Eggman have teamed up and combined the powers of their magic and technologies respectively to conquer a world that's being fused together by the chaos emeralds and the power stars.

I'd say a platformer but honestly I can't think of a good way to combine their core gameplay. Better to focus on the characters, worldbuilding, and storytelling, which would be the draw of that sort of crossover anyway.