You're perfectly normal, yes. Everyone looks different down there, and if you're skinny it's perfectly natural to be a bit bony in that area. Vaginal lips vary wildly in appearance, but "flappy" is hardly an unusual description.
And like the above poster stated, in the very unlikely event that your bf would actually leave you over that, it wouldn't be because there's something wrong with YOU. Because he'd have to be a real creep to do that.
So don't worry, okay? And if he keeps making a thing of it, just tell him that it makes you feel self-conscious and bad, and that you would appreciate him being a bit more tactful c:
Don't worry, you are completely normal. Those parts of our bodies are not the prettiest, neither is your boyfriend's penis probably. But they're useful. Don't be ashamed of your body. I bet you look great! Everyone has their insecurities, but there's no need.. If a guy leaves you for this reason, he'll probably be alone his whole life. Because a guy like that doesn't deserve a nice girl/guy. But he probably thinks you're beautiful, otherwise you two wouldn't be together ;)