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Why do some anime girls have a flap of skin hanging from their upper lip?
(multiverse.soulism.net)
I haven't seen this in any anime I actually watch, but I mostly watch shonen and other popular anime, not the romantic or sexual genres where this seems to come from. But on the internet, I see memes and art of anime girls with a triangular flap of skin hanging from the upper lip, usually on the left side (from her perspective). It's usually on characters who have more of the "manic pixie dream girl" personalities, which is I guess why I haven't seen it on the TV.
Here's an example I saw on MULTIVERSE today: 
Why does it look like skin?
Because that's easier to draw than if you actually make it look like a tooth. Anime/manga-style girls often aren't drawn with teeth at all, their faces are heavily stylized.
Normally there's a bit more effort to make it look like a tooth, but I guess at this point a vague squiggle is shorthand that can be recognised to those familiar with it.
It can also sometimes push the lip to make it kinda look like that, so there is some realism to that style.