Is the word cardápio the same in Brazilian and European Portuguese?

No

While the fundamental meaning of the word is identical, there is a difference in typical usage. In Brazil, "cardápio" is the standard and most common term used to refer to a list of food and drinks in a restaurant. In Portugal, although "cardápio" is understood, the word "menu" is much more natural and widely used in everyday conversation and dining establishments.

Brazilian Portuguese Examples

1. Você pode me trazer o cardápio, por favor? (Can you bring me the menu, please?)
2. O cardápio de hoje é feijoada. (Today's menu is feijoada.)
3. Eu gosto de olhar o cardápio antes de pedir. (I like to look at the menu before ordering.)
4. O cardápio de bebidas é bem variado. (The drinks menu is very varied.)
5. O que tem no cardápio hoje? (What is on the menu today?)

European Portuguese Examples

1. Pode trazer-me o menu, por favor? (Can you bring me the menu, please?)
2. O menu de hoje é feijoada. (Today's menu is feijoada.)
3. Gosto de dar uma vista de olhos no menu antes de pedir. (I like to take a look at the menu before ordering.)
4. O menu de bebidas é bastante variado. (The drinks menu is quite varied.)
5. O que é que tem no menu hoje? (What is on the menu today?)