Oof, I can relate to this. Each side of my family made politically incorrect remarks about the ethnicity of the other side, from the time I was a child. I can relate to feeling like you have no choice but to laugh along, while underneath, there's this confusion: "Do they also see me in this unfavorable light, because I am one of these people?"
I'm sorry you feel insecure about your lips. Some of the most beautiful women I know have thin lips, and their lips don't detract from their beauty one bit. If the comment section here is any consolation, I hope it serves as proof that the features we dislike about ourselves usually don't even ring any kinds of bells in anyone else's mind; what I mean is, I'm guessing nobody sees anything wrong with your lips, just like apparently no one saw anything wrong with my nose. ;) May you find a place of more confidence too, Gig! :)