(I am not a lawyer, but I am reasonably well-read.)
Nintendo would have poor legal footing to take down this website. This website exists for the purpose of reporting on information related to the Pokemon franchise -- it isn't a substitute for the Pokemon franchise or any products associated with it. It uses copyrighted artwork, but various protections exist for the use of copyrighted works for the purpose that websites like Pokemon Database and Serebii use them. In particular, American fair use law (and adjacent fair dealing laws in other countries) gives concrete protection to websites like this one which use copyrighted works for "reporting", "research", etc.
Nintendo doesn't "own" information like the fact Squirtle learns Water Gun at level 3, so the body of the content on this website is absolutely clear of any legal problems.
This is not to say that Nintendo couldn't attempt to take this website down, but in light of the above (and the fact Pokemon Database probably generates more interest in their brand than it detracts), they would have no interest in doing that.
It is a lot harder to argue that Nintendo soundtracks uploaded to YouTube in their entirety are anything but a violation of copyright (as much as many people rightfully argue that the music should be available for people to listen to), and the use of Nintendo/The Pokemon Company's characters in fan games certainly doesn't fare any better. Websites like PokemonDB are not easily comparable.
I'll also clear some issues brought up in this thread:
- Whether or not Pokemaster makes money off this website is not totally irrelevant to the discussion of fair use, but it certainly isn't a deciding factor. The nature of the work is much more important.
- Nintendo almost certainly doesn't care about PokeBase and whether we allow questions about fan games.
- Discussing the legality of this website is not something Nintendo's customer support services will help you with.
