Unlike other Tales games you obtain Monster Data simply by encountering the monsters. There is no need for Magic Lenses which saves you a lot of effort when it comes to this trophy. There is a bit of leeway if you choose to do this while playing Milla's story, as there is a later part of the game where you face monsters exclusive to her side. Pair these with your sub events for certain enemies and you should have no problem reaching the 275 unique enemies for this trophy.
There are encounters in the arena that count towards this including doubles of Presa, Jiao, Wingul, Ivar and doubles of your whole team.
There are also a few sub events that offer unique enemies. These could range from standard enemies that are slightly different for sub event purposes or special enemies such as the Devil's Beasts or the Guardians in the Xailen Woods Temple.
Bacura's can be missed as they don't spawn all of the time and flee when you see them, simply use a dark bottle and they will instead run towards you.
Otherwise, simply fight every enemy you see and keep working towards completing the monster book! If you've saved enough grade from your first play you can keep all monster data for the second run.