Actually, Ash happens to be based off Red (who is the player character in the original games). Then, Pokémon Yellow is the same game with some features based off the anime. In that game, you obtain Pikachu, along with all three starters (just like Ash) and a Lapras in HGSS (or Eevee/Espeon in GSC). You're also given a PokéFlute so you have to encounter a Snorlax too. You become the Champion at the end of Pokémon Yellow. So at the end of GSC/HGSS, you are essentially fighting the character you played as, with the team of Pokémon mentioned above.
I know that paragraph probably makes no sense, but I'm too lazy to try to fix it up. It's so confusing to explain but at least I know what I mean. xD