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

Yes

The word "nenhum" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are generally more open and the final "u" is clearly articulated. In Continental Portuguese, unstressed vowels are often reduced or "swallowed," meaning the "e" is much more subtle and the "u" is much more clipped.

Brazilian Portuguese examples:

1. Não tem nenhum problema. (There is no problem.)
2. Eu não vi nenhum cara na festa. (I didn't see any guy at the party.)
3. Não sobrou nenhum pedaço de bolo. (There wasn't any piece of cake left.)
4. Nenhum de nós sabe a resposta. (None of us know the answer.)
5. Não encontrei nenhum celular no quarto. (I didn't find any cell phone in the room.)

Continental Portuguese examples:

1. Não há nenhum problema. (There is no problem.)
2. Não vi nenhum rapaz na festa. (I didn't see any young man at the party.)
3. Não sobrou nenhum pedaço de bolo. (There wasn't any piece of cake left.)
4. Nenhum de nós sabe a resposta. (None of us know the answer.)
5. Não encontrei nenhum telemóvel no quarto. (I didn't find any mobile phone in the room.)