Menu vs carte
So, my understanding is menu is carte in French. But duolingo teaches menu. Why?
Well I think it depends on the context and how you say things. If you just sat down and ask "le menu", it can mean that you want the list of everything served here. So it's not entirely wrong, though you're write we mainly use "la carte" as "un menu" is often the combination of all the part of the meal for a cheaper price.
From my experience of Duolingo, most courses try to make you learn "transparent" words and god knows there are a lot between French and English. You can still be understood using menu instead of carte so that's okay somehow. Duolingo is mostly here to give you small basis, once you will have heard waiters telling you several times "voici la carte", you'll automatically adapt your knowledge and that's how learning languages works.