The more recent Persona games combine dungeon crawling with dating sim and visual novel elements. Fight demons, befriend all sorts of people, date a cute girl and don't let your grades slip. That may sound pretty lame, but it's totally amazing thanks to clever gameplay mechanics, extremely likable characters and writing that can be suprisingly mature.
I have to say this is one of the most apt descriptions of P4 I've ever read. Kudos to you Tomara, you really captured the qualities of P4 in a clear and succinct manner. :-)