There are a ton of pets in the Pokémon franchise.
For example, meowth, lillipup, and purrloin.
Almost all of the dog and cat looking Pokémon are pets in the Pokémon franchise.
There are also a lot of bird based Pokémon that are viable as pets, as we have a old woman with a ton of rookidees in her house in Pokémon Sword and Shield.
So based on the facts, I'll list them out.
Long list incoming:
Meowth, lilipup, purrloin, rookidee, chatot, galarian meowth, delcatty, skitty, shinx, espurr, eevee, furfrou, growlithe, yamper, rockruff, vulpix, zorua, herdier, electrike, manectric, snubull, boltund, togekiss, cubchoo, teddiursa, minccino, cincinno,
alolan meowth (somewhat a pet), luvdisk (seen in Pokemon sun and moon in professor kukui's house, not much of a pet though.)