To directly answer the OP, the game is incredibly charming in its presentation. The user interface elements are all designed extremely well and are bright and fun and thoughtfully put together. Same goes for most of the game. The brightness is tinged with the occasional classical SMT darkness and mystery, which adds a lot of intrigue and is used to drive the story forward. I really think the charm takes people off guard and sucks them in. The game itself is great, in a ton of ways.
From a post-first-impression perspective, one of the best parts about Persona 4 is how integrated the game is. Which is why, when people complain about particular segments of the game (the dungeons, the combat, the social links, etc) it makes me kinda concerned that they missed (or intentionally ignored) something. You're never doing just one thing in the game. When you're in a dungeon, you're fusing and you're exploring the dungeon and you're fighting shadows. The more you fight, the more personas you get, the more you can fuse, the more new skills and skill-sets you can try. When you get out of the dungeon, you can sell items, buy new ones, improve social links (which improves your ability to fuse), fuse more, etc etc.
There's TONS of content in the game, and I don't just mean big large story segments. The game itself lasts a very long time, but it's also filled with tons of small details. Conversations and interactions with things that people may never find in half a dozen playthroughs.
The game sucks you in with its presentation and keeps you ensnared by connecting everything together. It's a high quality product that feels like it was genuinely made with love. It's enjoyable.
I definitely feel this way about the game. It has a similiar appeal to an elder scrolls game that way(Granted they are completely different games).
Honestly, I wish I knew. I've never quite been able to understand it.
It's not the waifus. Gajllions of animu waifu games come out every year, and they're almost all incredibly niche.
It's not the dungeon gameplay. Other SMT titles use similar but better versions of the same combat systems, but are less popular.
It's not the story or writing. Well, actually, it seems that it is - the game receives a lot of praise for these things. I just don't understand why. Nothing about the narrative stands out to me compared to other games or anime. It's very... normal.
It's also very similar to Persona 3, but much, much more popular. This is maybe the most confusing point - it is somehwat improved, but still.
I can only assume it's some transient, magical combination of these things.
I feel like it is the combo. Persona doesn't seem to excel at any one aspect of it that makes people like it. As for the Waifu point, to be fair, pretty much all WRPGs nowadays have a romance aspect to them, so it's not an uncommon mechanic at all, though.
I also notice there are a lot of people who dislike it because of the slow intro. Then there are people who dislike it for the reasons that I expected my shooter friend to dislike it...