You do level up when you defeat them yes. Only a bit later on in the game can you capture them and make them a familiar. But first the monster has to fall in love with you after you beat them. JRPG's... gotta love them.
Haven't tried capturing two of the same though. I like my variety.
PS+ games confirmed for February.
http://mygaming.co.za/news/ps4/61565...us-update.html
That's what she said.
The Dork Knight.