This comes up often, and here's the general answer.
In this specific case, menu can be viewed either physically, printed on paper to read, i.e,
- 2-Dimensional, in which case on is used
- It's right there on the menu; you can't miss it.
or informationally, like a list, or a report; this a Container metaphor, i.e,
- 3-Dimensional, in which case in is used.
- It's hard to resist the Armenian dishes in the menu.
Quite often it makes no difference, because the information that's in the menu is also printed on the menu.
This alternation is true of any physical noun that can be interpreted as information, like story, picture, book, article, paper, report, list, plans, etc. These are collectively called "Picture Nouns" and they have very complex syntax in English.