hmm a lot of people have compare to pokemon... i'm curious!
The comparision to Pokemon is due to the various Personas that you can collect and use in combat. In P3 and 4, your other party members use only one Persona, which essentially has them growing with a standard level up system (gaining various abilities at various levels). But the protagonist is capable of using over 100 different types of Personas. In the case of 3 and 4, you have a chance of picking up a new persona if you meet certain conditions in battle. The persona you have equipped will determine all of your stats and abilities except for your HP and SP. The personas each level up an gain new moves individually, just like your party members do.
In addition, you can fuse two or more Personas together to make a new persona, which is usually more powerful and has inherited abilities from the seed persona, along with learning the abilities that it learns on it's own.
This all also ties into the life sim aspect of the game. Each Persona is part of a family named after one of the Arcanas of Tarot Cards (such as Empress, Magician, Star, and Devil). And the same holds true with the various people you can make friends or get into a relationship with (Your sports club friend might be Chariot, while the lonely little girl in the park could be Hanged Man). As you meet with and form bonds with these people, your ranks in these Arcana's increase. And as those ranks increase, you get an exp boost in the corresponding Persona family. So if you know that you've been hanging out with that Sports Club friend a lot, you could take advantage of that by fusing more Chariot Personas so you can enjoy the level boost that they'd get from your bond.
This
Atlus video showing off fusion in Persona 4 should help to paint a better picture.
The gameplay basically becomes a balancing act of hanging out with your friends to get significant strength gains (and flesh out their story along the way), and then going into the dungeons to progress with the plot and get stronger through there.
Granted, this is P3 and 4. There's always the chance that things may get changed up in P5 to an extent, so we'll have to see.