Nintendo continues its fight against gamers

[EeveeBailey]

No, it does not fall under fair use. Pokemon Essentials uses Nintendo IP and they have every legal right to demand it be taken down. Fair use only applies to parody and very small selections of an IP for criticism. Even fanfiction is not legally protected.

According to the link that Mika posted though, transformative works have fallen under the category of Fair Use, particularly in several U.S. Supreme Court cases:

The first factor is "the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes." To justify the use as fair, one must demonstrate how it either advances knowledge or the progress of the arts through the addition of something new.... A key consideration in recent fair use cases is the extent to which the use is transformative. In the 1994 decision Campbell v. Acuff-Rose Music Inc,[11] the U.S. Supreme Court held that when the purpose of the use is transformative, this makes the first factor more likely to favor fair use.[12] Before the Campbell decision, federal Judge Pierre Leval argued that transformativeness is central to the fair use analysis in his 1990 article, Toward a Fair Use Standard.[9] Blanch v. Koons is another example of a fair use case that focused on transformativeness. In 2006, Jeff Koons used a photograph taken by commercial photographer Andrea Blanch in a collage painting.[13] Koons appropriated a central portion of an advertisement she had been commissioned to shoot for a magazine. Koons prevailed in part because his use was found transformative under the first fair use factor.

If there's no commercial gain, just art for the sake of art -- or education, as some could argue that Pokemon Essentials would be (archiving sprites, teaching programming skills) -- such as would be the case with fanart, fanfic, and even fangames, then by the precedent set in the Supreme Court, it would be Fair Use.

That is false. You can't just steal intellectual property and claim you transformed it. Otherwise putting a Pikachu on a T-shirt and selling it would be fair use. It's not. Fair Use is not a magic wand.

Transformative usually means a whole new medium, or a transformation that makes a statement (parody), or using intellectual property as a new type of building block (sampling a song for a remix). The purpose of the IP will have to change. We went from a game to a game

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Again though, cases are skewed toward Fair Use primarily when transformative and not used for commercial gain.  You're right, putting a Pikachu on a T-shirt and selling it would not be considered Fair Use because you're profiting on merchandise.  But to draw Pikachu and screenprint a T-shirt for your own use?  Something completely different, even though you're still using Nintendo's IP -- it's both transformative and not being done to make money.

I would still argue that most fangames would fall under Fair Use by using your very example of how sampling a song for a remix would be using IP as a building block.  Yes, we went from one game to another game... but that remix went from one song to another song.  Perhaps I'm missing something, but I fail to see where the difference lies in your comparison.