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Of the starters, the only lines to maintain a degree of passivity are Squirtle and Turtwig (being based on tortoises), with some like Bulbasaur and Chikorita being considered docile with no animal basis. Tepig can make a good pet, though boars are very dangerous animals in the wild. I've actually taken care of a python like Serperior, they're difficult but not impossible to raise. The Oshawott line (otters and sea lions) can be trained as good pets, but can also be dangerous.
Pokemon like Charizard (a dragon), Typhlosion (a badger or a wolverine), and Feraligatr (a fucking alligator) can be ruled right out. Pokemon like Sceptile and Swampert are dangerous in their own right as being territorial. While helpful domesticated monkeys exist, everyone knows an ape like Infernape is fucking terrifying. Pokemon like Torchic and its line are pure fighting Pokemon, and thus cannot be considered 'good' pets.
All in all, I would probably say Turtwig, and thus, Torterra. It's herbivorous as well as photosynthesizes, and is famous for its gentle nature with other animals. The worst starter for a pet would probably be Charizard.