What does the phrase "fashion doctor" mean here? A sarcastic term that refers to a fashion nerd? Or a compliment for those who know fashion?

Rebecca

The phrase "fashion doctor" in this context is used to mean that she thinks the lipstick is fashionable, and if she uses it, will make her more fashionable. It also indicates that the lipstick was meant for her to use, since she relates it to a prescription that was ordered by a doctor. This is a very casual expression.