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

Yes

The word "serem" is the third-person plural of the personal infinitive of the verb ser (to be). It is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference is in pronunciation: in Brazilian Portuguese, the vowels are more clearly articulated and "open," whereas in Continental Portuguese, unstressed vowels tend to be reduced or almost silent, making the word sound more like "s'rem."

Brazilian Portuguese usage

1. É importante eles serem honestos. (It is important that they are honest.)
2. Para eles serem bem-sucedidos, precisam estudar muito. (For them to be successful, they need to study a lot.)
3. Eles têm o direito de serem ouvidos pela diretoria. (They have the right to be heard by the board.)
4. O problema é eles serem muito distraídos durante a aula. (The problem is that they are very distracted during class.)
5. Sem eles serem tão legais, a festa não seria divertida. (Without them being so cool, the party wouldn't be fun.)

Continental Portuguese usage

1. É fulcral eles serem honestos. (It is crucial that they are honest.)
2. Para eles serem bem-sucedidos, têm de estudar muito. (For them to be successful, they must study a lot.)
3. Eles têm o direito de serem ouvidos pela direção. (They have the right to be heard by the management.)
4. O problema é eles serem muito desatentos durante a aula. (The problem is that they are very inattentive during class.)
5. Sem eles serem tão prestáveis, a festa não seria divertida. (Without them being so helpful, the party wouldn't be fun.)