First, under no circumstances would I gate iconic Mario abilities behind classes and especially not prestige classes.
I would do powerups two different ways, as both magic items and feat chains-- so anyone can pick up a Fire Flower, but some characters don't have to.
Probably take a limited selection of Edges and Talents from Rogue Genius Guides' Guide to the Talented X series, especially Monks, and just hand them out-- everyone gets a ki pool with the ability to boost dashes and jumps, which characters can choose to specialize in or not.
I've done the Mushroom Kingdom as both all-human (from Earth) and as diverse natives. Have a preference for the latter.