The fundamental problem with this question, and every question phrased like it, is this: There's a difference between something for kids, that is something that is directed at a very young audience, and something that is only for kids, that is something that doesn't sustain much interest for anyone who's matured at all.
It's the difference between Barney and Sesame Street, between the first and second Teenage Mutant Ninja Turtles movies, between Harry Potter and forgettable young adult novels, between the best of Pixar and Disney and the bottom of the barrel of Saturday morning kid's cartoons, and between Calvin and Hobbes and inferior strips. Isaac Asimov wrote about writing for young people in Gold, and he pointed out that many of the apparent differences between literature that works for kids and the literature that doesn't are fictive. He points out that children swear, more closely emulating South Park's children than our beatific imagination of innocence, and that children can engage with material that can be deep, moving and challenging. Any kid who came up in the 80s and 90s and watched Ghostbusters, Total Recall, Terminator, Terminator 2 and Robocop on VHS knows this fact. Children can absorb some tough stuff, and adults can appreciate childish whimsy and wonder.
What makes any property, from a video game to a film, work is having good fundamentals and some kind of payoff. Ghostbusters has lots of snarky dialog that you won't get as a kid, but the idea of running around with science gadgets catching ghosts is fun, and while Dana being abducted is pretty freaky, the stakes are light enough that a lot of kids can handle the scarier aspects. Robocop is full of satire and incredible violence that one can appreciate as an adult, as well as some authentically heart-wrenching moments where we realize that this father has lost his family and is trying to regain his humanity, but as a kid it's possible to appreciate a robotic cop getting revenge on a bunch of Dicks who shot him. Ironically, I find myself affected viscerally more as an adult by these works than I remember being affected as a child: Terminator 2 is the only movie that I remember getting the impact of as a child, and I suspect that's because the end of the movie is the tragedy of a son losing his father, which is something that a child can by his very nature understand. The rest of the adult classics that I watched as a child of the 90s took time for me to truly appreciate, and see that they're not actually all that fun or lighthearted under the surface.
Pokemon obviously works for kids. Does it work for adults?
Pokemon's challenge is incredibly low, even for its target audience. I played much more difficult games as a child. But the mechanics are rich enough that even someone playing very casually can fall deeply into trying to figure out the ideal team that would deal with all appropriate enemies.
And the core fantasy, going out and catching an array of iconic monsters, appeals to a childlike sense that we could find anything, even literal gods, in the grass. Cute animals make us feel like we are collecting an array of pets (without the reality of having to feed, play with, clean up after and care for all those cute animals that tends to happen if you try to adopt a pig and a dog and a cat and a snake in the real world, not to mention the real heartbreak when you lose your furry or scaly friend). Shigeru Miyamoto said that Zelda evoked the feeling of moving through a massive, dangerous world and labyrinthine homes that a child has: "When I was a child, I went hiking and found a lake. It was quite a surprise for me to stumble upon it. When I traveled around the country without a map, trying to find my way, stumbling on amazing things as I went, I realized how it felt to go on an adventure like this." Pokemon hits the same sweet spot in the brain: it appeals to the idea of going on a great globe-trotting adventure, while crucially never being that far from Mom and home. And as adults, our brains are still capable of evoking that same sense that we remember as children.
The fact of the hardcore Pokemon community, the speed-runners and those who compete in duels, indicates that the game is mechanically deep and well-balanced enough to sustain deep inspection of its systems. Anyone who's designed a board game, let alone a video game, can tell you that this isn't easy: most ideas are more like Tic-Tac-Toe than Chess, becoming degenerate and simplistic rapidly. Instead, Pokemon has had an evolving metagame.
Now, it's true that most adults who have never played or encountered Pokemon aren't likely to feel the hooks of the game sink in. And it is also true that most children growing up leave behind Pokemon: there's just not enough there for casual players going through the campaign. I think Nintendo really would do themselves a service if they either created an edition for adults or created a difficulty level more appropriate for adults (also ratcheting down some of the grinding). Pokemon is actually not as good as The Incredibles or Up: uninitiated adults by and large won't be engrossed, and a large portion of the target audience will dissipate as they grow up, moving onto bigger and better things. (I suspect this is a big part of their logic behind Pokemon Go: their design is so masterful that adults can enjoy collecting Pokemon).
Still, no one can say that your main-line Pokemon game isn't going to give you an incredible amount of simple, easy entertainment, and it's trivial for an adult to figure out a self-imposed challenge. Even for adults, a Pokemon purchase is a pretty wise investment for hours-to-dollars.
In conclusion: Like anything that resonates at the level that Pokemon does, the appeals go much deeper than superficial cuteness or mere nostalgia. Nintendo hasn't been the only video game designer from the 1980s to remain as one of the big console producers by accident: they know how to design a game, and Pokemon shows it.