They're like pets, yeah, but I don't mind having so many. Or getting even more, I guess, since I keep doing that too.

But you can only have six with you, the rest of them go to some weird PC system, that you can access in the PokeCenter. They're completely safe there though, just like in pokéballs, and then you can go to those Centers to switch between the pokemon in your current team and the pokemon in that -- system.