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

No.

While the spelling and basic meaning of "em" are identical in both varieties, there is a difference in typical usage regarding motion and destination. In Brazilian Portuguese, it is very common in colloquial speech to use "em" (and its contractions like no or na) to indicate a destination (e.g., "Vou no shopping"). In Continental Portuguese, the preposition "a" (and its contractions like ao or à) is used to indicate movement toward a place, while "em" is used to indicate being physically located inside a place. Additionally, there is a difference in pronunciation: the nasalized vowel in Brazilian Portuguese is typically more open and resonant, whereas in Continental Portuguese, it tends to be more closed or subtle.

Brazilian Portuguese usage:

1. Eu vou no shopping. (I am going to the mall.)
2. Eu estou em casa. (I am at home.) *3. Ele mora em Minas Gerais. (He lives in Minas Gerais.)
3. O copo está em cima da mesa. (The glass is on top of the table.)
4. Tudo aconteceu em um instante. (Everything happened in an instant.)

Continental Portuguese usage:

1. Eu vou ao shopping. (I am going to the mall.)
2. Eu estou em casa. (I am at home.)
3. Ele mora em Lisboa. (He lives in Lisbon.)
4. O copo está em cima da mesa. (The glass is on top of the table.)
5. Tudo aconteceu num instante. (Everything happened in an instant.)