So, aside from:
-Pokmon
-Digimon
-Monster Rancher
-Shin Megami Tensei / Persona
-Dragon Quest Monsters
-Yo-Kai Watch
(forgot this one)
I just beat Nexomon: Extinction this evening, and while I had fun with it and its predecessor, I wouldn't call them my favorite. They're both rather shallow when it comes to their battle system (being just a poor man's Pokmon, but with less types/elements and no dual-types), and the monster designs are all over the place.
I need to get back into Monster Sanctuary, though. This one is actually much different from Pokmon, being a Metroidvania with a different battle system that relies on the player chaining the right moves together to deal massive damage, and defend against the opponents doing the same to them.
Another one I played years ago was Tales of Symphonia: Dawn of the New World, which really shouldn't have been a monster-collector at all (one of the game's many problems). Especially when those monsters are competing for four battle spaces with the party members from the first game, and I believe Emil always has to be in the party so you really only have three spots.
"Be good to yourself, because everyone else in the world is probably out to get you."
- Dr. Harleen Quinzel