Ummmm, they could just change the positioning of the targets on the level to tailor to a particular character. (ie: Pikachu has a two targets atop each other that is easily hit by lightning, but that is not the orientation for a different character...)
The stage's overall layout could be pretty generic.... and each character's starting position on that stage does not have to be the same. For example, there could be a wall jumping U in the stage and a good wall jumping character might actually start off in that U (Think young link in SSBM). Now pikachu might not start off in that U but there might still be a target in it (which one would think is best hit with lightning).