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

Yes.

The word "dormir" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, vowels tend to be more open and clearly articulated, and the rhythm is syllable-timed, which gives the word a more "musical" or elongated sound. In Continental Portuguese, vowels are often reduced or "swallowed" (for example, the "o" in "dormir" often sounds more like a "u"), and the rhythm is stress-timed, making the word sound more clipped and shortened. Additionally, the pronunciation of the final "r" varies significantly by region in Brazil (ranging from retroflex to aspirated), whereas in Portugal, it is generally more subtle.

Brazilian Portuguese

1. Eu vou dormir agora. (I am going to sleep now.)
2. Ele está dormindo muito. (He is sleeping a lot.)
3. A gente precisa dormir. (We need to sleep.)
4. Você consegue dormir com esse barulho? (Can you sleep with that noise?)
5. Eu adoro dormir até tarde. (I love to sleep late.)

Continental Portuguese

1. Eu vou dormir agora. (I am going to sleep now.)
2. Ele está a dormir muito. (He is sleeping a lot.)
3. Nós precisamos de dormir. (We need to sleep.)
4. Tu consegues dormir com este barulho? (Can you sleep with this noise?)
5. Eu adoro dormir até tarde. (I love to sleep late.)