In general, this entire area of game development is filled with uncertainty. What is considered an "homage" or "fanfiction" and what is considered downright theft? Games are often abandoned due to multiple reasons, and they are not at all limited to simply being failures, and they could potentially come back to life, which would cause problems if somebody was making a copy.
Big Brother was one of the most popular games on this site, and it was abandoned anyway, due to patreon's new guidelines. Dark Silver is trying his hands on another game, but he is not having much success with his audience, to put it mildly. If this continues, he might decide to just switch back to BB (personally I think he will just quit for good) and find a way to make it work, without patreon getting in the way.
Another example is "Ethan's Legacy" by Vengeancexxx. Many people really liked that game, so when Vengeancexxx announced that he was quitting for an unknown time-length, due to personal reasons, many players wanted to continue the game. Now it has been many, many months, and Vengeance has suddenly popped up again. He might/might not intend to continue Ethan's legacy, who knows. The point is: If somebody had decided to take his game, characters and story, then Vengeance would have to compete against his own game. I find it unlikely that he would let that happen without objections, especially if money is involved, or if the game is moving in a direction, which conflicts with the creator's original vision.
I would say try to steer clear of this area, if possible, as it offers many pitfalls and requires very delicate footsteps. If you get consent from the original dev to continue his work, then I would say go for it. However, if you can't get an answer from him, or the answer is simply a "no", then I would say that it is somewhat disrespectful to just hijack his work and use it for your personal pleasure.
Of course this entire thing is subjective, and every case is different from another. Just remember that if a game provided you with so much happiness that you feel that you simply must have more, then you risk destroying what an awesome thing you had at the beginning, especially if you don't have the original creator's blessing, or simply lack the ability to use this content in a good way.
A good example of this is the new Star Wars movies. Disney bought the rights from George Lucas and then basically told him to fuck off, and they wouldn't listen to any of his ideas for the new movies. This resulted in episode 7 & 8. Episode 7 was fairly well recieved, at least in the beginning, but it wasn't really doing anything new. It simply rehashed previous movies and mixed it up a little. It was clear that Disney was afraid to try any drastic changes, but that was also the problem with the movie: It wasn't really their work. Episode 8 suffered from many of the same re-uses of elements from previous movies, but also tried adding some of Disney's own artistic ideas, which ended up exploding in their faces.
I will stop, as this is not a post about me ranting about star wars, but the general principle is the same: When you try to continue the work on something done by another, you risk much more than just facing copyright issues. You risk betraying the thing that you once loved, and might end up having people dislike you, if they feel you are not doing the original content justice. And even if you succeed, what have you really accomplished? You would have built an amazing house, on a foundation, laid by another, without whom your house would never have been a reality.
In my eyes, it is simply better to make up your own stories and do your own thing. If you like something, let yourself be inspired by it. Don't try to imitate something, but strive to surpass it instead. Have faith in yourself. Who is to say that your own idea wouldn't be able to blow our minds even more than the other game you enjoyed?
Just my 2 cents.