If you like the idea of Pokemon but would rather your summoned monsters have a harder edge (or at least more spikes and pointy bits), then you might be interested in Armored God, a mobile MMORPG that launched some time last week on Android that will let players summon and command monsters to fight alongside them.
The blurb for the game on the Google Play store sounds a bit like the opening of a fantasy novel written by a very angsty young man.
“Mankind is battling the power of evil since the day of the fallen angel. In the world full of evil and chaos, God Create the mighty Armor for the chosen heroes to bring salvation to the dark world. Epic dungeons and raids await you, slay the Titan and conquer the dark realm.”
Narrative cheese aside, Armored God seems to have a number of features going for it, including the aforementioned monster summoning, the ability to ride monsters into battle, raid and dungeon content, and team-based PvP. It also is apparently an autoplay MMORPG, which is either convenient or a non-starter for you, so bear that in mind.