Pokémon are magical
Personally, I would go with a Warlock (Pact of the Chain) as my base for the template, but instead of the Warlock spell list, I would have him "summon" Pokémon. Doing it this way would allow him to bring extra-planar monsters (Pokémon in this case) into the world of Forgotten Realms. As he got higher levels, he would get access to more powerful Pokémon he could summon. You could even make a Poke Ball be his arcane focus.
I would let him have his Pact of the Chain familiar be a Pokémon of some sort (not too powerful) that would be the one that always stays out and make Cantrips be cast through that Pokémon.
Many of the spells in D&D could easily be ported to being produced by a summoned Pokémon that shows up, does their effect, and then goes away (like in Smash Bros.), that way you could even ease this player into being able to be more comfortable playing a standard character in the future maybe.
The party role they would most easily fulfil would be Arcane Blaster, but could also be a Utility Caster. Most of what Pokémon can do most closely resembles magic in my mind.
Are there rules for this?
Yes, there are. There is a section in the DMG that begins on p. 287 about Modifying a Class. The section even suggests modifying a class's spell list as one of the easier ways to alter the class. It does caution about modifying the Warlock's spell list in this section, but it doesn't mean that you shouldn't do it, just that you need to pay attention to what you would be allowing them to do and how often.
This section of the DMG also covers Substituting Class Features and Creating New Class Options.
I personally feel like Pact Magic's ability to regain spell slots with a short rest transitions well to Pokémon (they need to rest before they can be summoned again). I also feel that since a Pokémon Trainer relies almost exclusively on their Pokémon's abilities rather than their own, modeling them after a casting class is more appropriate than modeling them after, say, a Ranger. Finally, I feel like letting them have a Pokémon familiar (their Pikachu that won't go in the ball) matches both genres rather well and treating all other Pokémon they have as temporary summons works well too, like "Charmander, go" = Burning Hands and then Charmander has to go back into the ball unless used again.
If you want to go with the idea that your player's Pokémon Trainer if from another world, you could explain that Poké Balls don't work the same way in the world of Forgotten realms. They can't be used to catch monsters, and any Pokémon summoned from them can't stay out very long. You could even explain concentration spells as if keeping certain Pokémon out of their balls is difficult for the trainer.
You could make custom Invocations that allow a Pokémon to "evolve" or something if you want to keep that more thematic.