Well, I'm currently rewriting an old Dutch Pokémon fanfiction where the rival started half a year earlier than the protoganist. Both started with a Bulbasaur, but rival's Bulbasaur evolved into Ivysaur before the protoganist starts his journey. So I believe the type doesn't have to be the only disadvantage/advantage you can give your characters.
Also in my fic, the characters can choose, WHICH specimen they will travel with. Do they want the advantage of a strong and experienced Bulbasaur which can evolve into Ivysaur any moment, but with an authority problem or a Bulbasaur who is quite young, doesn't know many advanced moves, but is easier to raise and turns out more loyal towards his or her trainer?