While this game doesn't focus solely on monster collecting, it's still an important aspect of the game.
In Kingdom Hearts Dream Drop Distance, instead of running around various Disney worlds with characters Donald and Goofy, you explore select worlds with new creatures called Dream Eaters. Friendly dream eaters are called spirits, while the enemy dream eaters you fight are called nightmares.
Spirits are created by combining Dream Fragments together with an item or spell, either by trial and error or using recipes found throughout the game. The player characters can also link with Spirits to perform unique attacks. You can even pet and play with your Dream Eaters which will help them gain new abilities, similar to Pokémon-Amie, although surprisingly this game did it first.
Although not a monster collecting RPG at its heart, Kingdom Hearts Dream Drop Distance can still be played as an alternative to Pokémon if you focus more on the spirit creating aspect, and you get the additional benefit of getting to visit several amusing Disney-based worlds on the way.