^Dang man, that's deep.
I think Pokemon is so addicting because it's a fantasy world that allows YOU, the player, to put themselves into a world that's not too farfetched from our own (so your suspension of disbelief doesn't have to be as large). You're in a world full of monsters that resemble known creatures and unknown creatures. It's like Where the Wild Things Are, but with more than one or two monster archetypes.
Each creature appeals to someone, unless you just hate Pokemon with so much passion that you won't even look through a listing to find the one that does stand out. Some people like certain Pokemon because it reminds them of an animal they like. Some like them because they're cute, other because they're weird, others because they're extraterrestrial like, and others because they look like they mean business. I don't know another series that gives you that much variety outside maybe Digimon (or another monster game) that has that broad appeal of monsters and that also puts you in the spotlight.
Yes, we have people who imagine what Red, Green, Gold, Black, Orange, I dunno whoever do during the journeys, but it's not like playing a Mario RPG where you know for a fact you're helping Mario reach his goal. It's hard to put yourself in Mario's shoes. More often than not, though, we put ourselves into those avatars. Would Red realistically have a team consisting of Venusaur, two Glitched Mews, Pidgeot and Nidorina (based on a team I once had just to break the game and find as many data destroying malicious glitches as possible)? I doubt it. Would I? No, but it's more likely. A more realistic team wouldn't have glitched Mews, but you see what I'm saying. And the same applies to everyone. We're all given an option to customize to our heart's content.
And the beauty of Pokemon is that, despite the fact that the games limit you to a specific profession in order to conclude the game's campaign, reasonably speaking, there are a lot of Pokemon based professions you can take on, some of them taking a pacifist direction. Who hasn't imagined being a Pokemon [insert your favorite profession] at some time?
Pokemon is a series that transcends what lies within a circuit board, and it enriches our imaginations to a degree. Other series have fans that love to imagine different scenarios for the characters in question, too, but I don't think anything else, or most anything else puts you in the position of what you would do beyond the scope of the game world.
That, and the Pokemon themselves are only templates with a few rules. That's what makes the Mystery Dungeon series fun. Every Pokemon is a species, not a character, meaning you can have as much fun with one in particular as you want, without feeling like you're messing with canon. It's like making stories using cats, as there is no one cat (or they aren't all acting on a hive mind). I also don't know too many series that let you get away with that, unless you focus on the mooks or something.