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Most of the starters in the games are just a coincidence of the situation the protagonist finds themselves in. Only Juniper says that the starters she gives out are ones that she gives to everyone. Without that, though, we have also seen someone catch their own starter with Wally, so there are other avenues, and they don't have to be the three we receive. It is unclear if they keep their starter, that seems to depend on the person. Only LGPE has portrayed the starters as actually being a part of the ecosystem. Gamefreak has said before that they make the region first, and then the pokemon that fit in with it. Supposedly, even the starters are supposed to feel as if they belong to the region. As for being rare, in LGPE, all their natural locations are rare.
For other media, this is very much the same. They aren't the only starters, although apparently, in the anime, it's part of the professor's job to distribute only them. Their importance still depends on the person, and their population is unclear, although for the most part, they are treated as if they were ordinary pokemon.
