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

Yes

The spelling, meaning, and grammatical function of "dormem" are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the unstressed "e" is usually more audible (though still somewhat reduced), whereas in Continental Portuguese, unstressed vowels undergo significant reduction, making the "e" sound nearly silent or extremely closed.

Brazilian Portuguese

1. Os cachorros dormem no sofá. (The dogs sleep on the sofa.)
2. As crianças dormem cedo todas as noites. (The children sleep early every night.)
3. Os bebês dormem muito durante o dia. (The babies sleep a lot during the day.)
4. Eles dormem sempre na sala. (They always sleep in the living room.)
5. As pessoas dormem melhor no escuro. (People sleep better in the dark.)

Continental Portuguese

1. Os cães dormem no sofá. (The dogs sleep on the sofa.)
2. As crianças dormem cedo todas as noites. (The children sleep early every night.)
3. Os bebés dormem muito durante o dia. (The babies sleep a lot during the day.)
4. Eles dormem sempre na sala. (They always sleep in the living room.)
5. As pessoas dormem melhor no escuro. (People sleep better in the dark.)