Depends on the context like everyone says. Example: Q: Would you like to have a bite (of the cake) ? A: Thanks, (but) I'm not hungry

It does seem depend on context. My best guess is that it's trying to convey a certain amount of sass or attitude, but without a picture or graphic to give further info it's basically open to interpretation.

Were there any pictures or symbols on the mug to give us a clue? To understand it properly, we will need to work out what the implied content is. So it is "Thanks (but) I'm not ..." Perhaps, "Thanks but I'm not available" or "Thanks but I'm not interested in you" or "Thanks but I'm not looking for anyone like you."