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

Yes

The word "nem" is identical in Brazilian and Continental Portuguese in terms of its meaning (functioning as "neither," "nor," or "not even"), its grammar, and its spelling. The only difference is pronunciation; Continental Portuguese features significant vowel reduction, meaning unstressed vowels are often pronounced more closed or even swallowed, whereas Brazilian Portuguese tends to maintain more open and distinct vowel sounds.

Brazilian Portuguese Examples

1. Eu nem vi você chegar. (I didn't even see you arrive.)
2. Não gosto nem de frio, nem de calor. (I don't like either cold or heat.)
3. Ele nem sequer respondeu minha mensagem. (He didn't even respond to my message.)
4. Nem tente fazer isso sozinho. (Don't even try to do this alone.)
5. Não tenho nem um centavo no bolso. (I don't even have one cent in my pocket.)

Continental Portuguese Examples

1. Eu nem vi que chegaste. (I didn't even see that you arrived.)
2. Não gosto nem de frio, nem de calor. (I don't like either cold or heat.)
3. Ele nem sequer respondeu à minha mensagem. (He didn't even respond to my message.)
4. Nem tentes fazer isso sozinho. (Don't even try to do this alone.)
5. Não tenho nem um cêntimo no bolso. (I don't even have one cent in my pocket.)