The phrase "the same menu" is correct and usable in written English.
You could use this phrase to refer to a menu that has not changed, or that is the same as another menu, such as in the sentence "The restaurant has been serving the same menu for the past 10 years."
Exact(59)
Most important, it offers the same menu.
The same menu includes "lighter fare".
We've never done the same menu twice".
What are they doing on the same menu?
And do you have the same menu every night?
THE BILL The same menu is served all day.
All from the same menu, using the same remote.
Both Tayyabs and New Tayyabs served the same menu.
But not, one hopes, all off the same menu.
A lush garden terrace has the same menu.
THE BILL The same menu is available for lunch and dinner.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com