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

Yes

The word "xadrez" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The difference lies exclusively in pronunciation. In Brazilian Portuguese, the vowels are generally more open, and the final "e" is typically pronounced clearly. In Continental Portuguese, unstressed vowels are significantly reduced; the "a" becomes more closed/centralized, and the final "e" is often shortened to a sound similar to a very light "i" or becomes nearly silent.

Brazilian Portuguese examples:

1. Eu adoro jogar xadrez com meus amigos. (I love playing chess with my friends.)
2. Ele comprou uma camisa xadrez nova. (He bought a new checkered shirt.)
3. O jogo de xadrez começou cedo. (The chess game started early.)
4. Eu gosto desse padrão xadrez. (I like that checkered pattern.)
5. Você sabe jogar xadrez? (Do you know how to play chess?)

Portuguese (Portugal) examples:

1. Eu adoro jogar xadrez com os meus amigos. (I love playing chess with my friends.)
2. Ele comprou uma camisa xadrez nova. (He bought a new checkered shirt.)
3. O jogo de xadrez começou cedo. (The chess game started early.)
4. Eu gosto deste padrão xadrez. (I like this checkered pattern.)
5. Tu sabes jogar xadrez? (Do you know how to play chess?)