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

Yes

The word "nosso" is identical in both Brazilian and European Portuguese regarding its spelling, grammar, and fundamental meaning ("our"). The only differences are phonetic. In Brazilian Portuguese, the final "o" is generally pronounced as a clear, rounded vowel. In European Portuguese, unstressed final vowels are often reduced or "devoiced," making the "o" sound more like a short, muffled "u" (/nuʃu/).

Brazilian Portuguese

1. Nosso cachorro é muito fofo. (Our dog is very cute.)
2. Este é o nosso carro. (This is our car.)
3. O nosso time ganhou o jogo. (Our team won the game.)
4. Adoro o nosso jeito de falar. (I love our way of speaking.)
5. Nosso plano deu certo. (Our plan worked out.)

European Portuguese

1. O nosso cão é muito fofo. (Our dog is very cute.)
2. Este é o nosso carro. (This is our car.)
3. O nosso grupo ganhou o jogo. (Our group won the game.)
4. Adoro o nosso modo de falar. (I love our way of speaking.)
5. O nosso plano correu bem. (Our plan went well.)