TLDR: There's no reason to play Pokemon online with other players outside of battles and trading. Same goes for most of Nintendo's other franchises, which focus on individual battles or game sessions, rather than persistent-world mechanics.
Pokemon isn't suitable for an MMO because there's no reason to play the main game with other players, on- or off-line. The only parts of Pokemon that support any online component are the battle mode and the Global Trade, which both make use of the internet to find suitable players. Otherwise, there's no point in walking around not!Japan while your buddy is on the other side of the in-game continent; what's to guarantee that you're playing in the same area and can progress together? The real multiplayer component of Pokemon comes from the metagame, the society and subculture it creates.
Mario and Zelda have both experimented with the multiplayer field: the most recent (and most successful) trek has been NSMBWii, which was more of a face-to-face thing, and before then there was the short-lived Four Swords franchise. In both of these cases, multiplayer was based on a cooperative/competitive mode of play, as opposed to the typical persistent-world approach for MMOs.
The only thing an online component would bring to Nintendo's multiplayer philosophy is player matchmaking, as they did with Pokemon's Global Trade System, SSBBrawl, and Mario Kart Wii, but it's general consensus Nintendo still lacks the serious technical chops to make synchronous online play feasible for most of their own games.