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

Yes

The word "vem" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. It is the third-person singular (he/she/it) or the second-person singular imperative (informal "you") of the verb vir (to come). The only difference is pronunciation: in Brazilian Portuguese, the nasalized vowel is typically more open and clearly articulated, whereas in European Portuguese, the vowel is often more closed or even reduced.

Brazilian Portuguese Examples

1. Vem aqui agora! (Come here now!)
2. Ele vem de ônibus para o trabalho. (He comes by bus to work.)
3. Ela vem com a gente amanhã. (She is coming with us tomorrow.)
4. O trem vem muito rápido. (The train comes very fast.)
5. Vem ver isso! (Come see this!)

Portuguese (Continental) Examples

1. Vem cá agora! (Come here now!)
2. Ele vem de autocarro para o trabalho. (He comes by bus to work.)
3. Ela vem connosco amanhã. (She is coming with us tomorrow.)
4. O comboio vem muito rápido. (The train comes very fast.)
5. Vem ver isto! (Come see this!)