I don't like Monsters, they can be quite tricky to manage.  Whether they grow stronger or weaker depends on the type of meat you feed them.  Obviously, the best meat is of monster bosses, such as Ashura.  Monsters totally change their appearance and attacks each time you feed them.  They are useful at first since they can instantly grow to a decent level of strength/defense, but soon the non-Monster members of your battle party grow stronger.  Then your Monster(s) isn't strong enough to beat a boss or survive through a dungeon.