@bgillisp is correct, and the USA is one of those countries where companies holding the rights to an IP need to pursue action because you cannot selectively enforce and failure to defend means losing the rights. Either the company goes after John Q. Public's one-man-studio fan game, or they can't stop a rival studio from infringing that IP later on.
And yes, all forms of copyright infringement such as Fan Art, Fan Fiction, Fan Games, etc. are illegal and Cease and Desist orders can be issued at any time to have that content removed with the threat of legal action. When the Internet was a lot smaller than it is today it was actually very risky for a site to even host content with infringing IPs and sites used to get shut down. These days that stuff is so prolific that no one could hope to stamp out everything, so precedent is that "small" IP infringements like Disney art on DeviantArt get overlooked unless someone important finds out or it's being sold commercially without rights. Sort of like old-time military officers in movies not strictly enforcing regulations with plausible deniability ("Put that booze away before I'm forced to notice it"). As long as the company holding the IP can make a reasonable claim that they were unaware the infringing image/fiction/game existed then they maintain rights as long as they take action once it has been brought to their attention.
So making Final Fantasy fan art is fine, but if it gets noticed too much and someone sends it to SquareEnix, then SquareEnix would be forced to issue a C&D and pull the content or else they risk losing IP rights to that material. And if they lie and say they didn't know about it but are ordered to turn over their emails then they can add perjury charges to their problems. So it's way easier to just go after the little guy if they step out of line.
Also worth noting is that Don Bluth very likely does not hold the rights to any of the characters in those movies anymore. The rights to those movies are probably owned by Warner Bros. or Universal or Disney. Just because Disney didn't publish it back in the day doesn't mean that they didn't buy it after the fact. And I know for a fact that those 3 absolutely do go after people for copyright and trademark violations.
Japan's massive "doujin" events like Comiket where tons of trademark infringing stuff is sold is only possible because Japan's IP law is massively less strict. And the fact that most anime and manga artists at the major studios all got their start doing fan works of popular characters means that the actual artists of the characters being copied strongly defend the right to make said fan works since it builds the future generations of artists. The anime/manga companies in Japan actually wind up being big players in lobbying against strict less since they value the symbiotic relationship with their works and fan works.
(fun fact: this was usually in the form of hentai, so the creators of many modern classic anime and manga got their start making pornographic material)
I can't recommend making a fan game since there are issues, even if most people wouldn't encounter them due to the vastness of the web. But if you were to be one of the unlucky ones to have your project noticed by the wrong people I'd hate for all of your work to get slapped with a C&D. It's really better to just make an original game. Make a character that is nearly identical if you want those exact character traits. It's not like any of the characters in the Don Bluth movies and Final Fantasy are even truly original in the first place anyway. As the saying goes "All that can be said has been said. There is nothing new under the sun."
