This is what’s called a “speech-act.” He makes his choice official by saying it. Speech-acts are in the present indicative. The classic example is the traditional wedding ceremony, which has several of them: “Do you take ...?”/“I do,” “With this ring, I thee wed,” and “I now pronounce you husband and wife.”
He (and his rivals) are saying this in the context of a sporting match, where it seems to be a rule of the game that trainers must announce the Pokémon they send into the arena. It’s therefore somewhat similar to players having to say, “Check!” in Chess, but even more like needing to say, “J’adoube” (“I adjust [it]”) to be able to touch a piece without moving it.
This could also have been phrased in the jussive mood. “Let Picachu battle!” would have been like “Let the games begin,” “Let there be light,” or “Red Rover, Red Rover, let Alice come over.”