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

Yes

The word "bairro" is identical in both Brazilian and Continental Portuguese regarding meaning, spelling, and grammar. The only difference is in pronunciation. In Brazil, the vowels are typically more open and the rhythm is more syllable-timed (each syllable receives more equal weight). In Portugal, the unstressed vowels are much more reduced or "swallowed," and the rhythm is more stress-timed, which can make the word sound shorter and more closed.

Brazilian Portuguese Examples:

1. Eu moro num bairro bem tranquilo. (I live in a very quiet neighborhood.)
2. O bairro está ficando muito movimentado. (The neighborhood is becoming very busy.) *3. Tem um parque muito bonito no meu bairro. (There is a very beautiful park in my neighborhood.)
3. Eu gosto muito de passear por esse bairro. (I like walking through this neighborhood very much.)
4. Esse bairro é muito legal. (This neighborhood is very cool.)

Continental Portuguese Examples:

1. Eu vivo num bairro bastante tranquilo. (I live in a quite a peaceful neighborhood.)
2. O bairro está a ficar muito movimentado. (The neighborhood is becoming very busy.)
3. Há um parque muito bonito no meu bairro. (There is a very beautiful park in my neighborhood.)
4. Gosto muito de passear por este bairro. (I like walking through this neighborhood very much.)
5. Este bairro é muito giro. (This neighborhood is very nice/pretty.)