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: 
Skin fang. Kinda like the lower lip having a dip to show drooling. Its a style that one artist did then everyone else liked it so kept it going. Skin fang supremacy BTW.