The closest thing we have to an anime character playable in Smash is Mewtwo, whose personality in the Japanese version of Melee was very closely based on the Japanese version of Pokémon: The First Movie. They also use Shadow Ball, which isn't a Psychic type move, but Mewtwo's anime movie incarnation is the first Pokémon to ever use the move Shadow Ball in any Pokémon media, so it's associated with Mewtwo in Smash.
Other Pokémon have elements that call back to the anime (Smash 4 and Ultimate Lucario use Lucario's VA from their movie, Smash Ultimate Greninja now uses the kunai the anime depicts them using attacks with instead of the Smash original water katanas) besides the anime trope of "their names are the only thing they can say", but Mewtwo is seemingly a special case.