I don't think it doesn't really matter as long as it is done right. But if you use "edgy" in the negative way, then for me it is something like this:
Most stories I perceive as edgy are often those, where the world is only there to make the MC look cool. Like there is no reason behind the state of the world. It's only so that the MC can suffer through it and come out on top, being the cool guy.
You often have that in anime, where the MC suffers on his own in the dungeon, on his hunt, on whatever and he becomes the cool and moody kid. And then he meets the harem member that lives in the same world, didn't suffer for a day, had a great good time in the same dungeon, and thinks the entire world is great.
It gets even worse, when there isn't even a real consequence in the story for it. He just suffers through it, doesn't really change after a level up, and goes on to live a happy life. This part of the world forgotten, until he meets that one noble that kicks him down. And of course, only him. Then he wants to go somewhere but the guards stop him. Of course, only him out of the 10,000 guests for the big event...
But to be fair, even a perfect dark story can be disturbed by stupid motivations or a weird sentences that make me laugh. But those "edgy moments" are often fine, if they don't appear too often...