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

Yes

The spelling, meaning, and grammatical function of "nada" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, vowels are generally more open and clearly articulated. In Continental Portuguese, unstressed vowels (such as the 'a' in "nada") undergo significant reduction, making them sound much more closed or even nearly silent.

Brazilian Portuguese

1. Eu não quero nada agora. (I don't want anything right now.)
2. Não tem nada na geladeira. (There is nothing in the refrigerator.)
3. Você não disse nada para ele. (You didn't say anything to him.)
4. Nada é impossível com esforço. (Nothing is impossible with effort.)
5. Ele não faz nada o dia inteiro. (He does nothing all day long.)

Continental Portuguese

1. Eu não quero nada agora. (I don't want anything right now.)
2. Não há nada no frigorífico. (There is nothing in the fridge.)
3. Tu não disseste nada a ele. (You didn't say anything to him.)
4. Nada é impossível com esforço. (Nothing is impossible with effort.)
5. Ele não faz nada o dia inteiro. (He does nothing all day long.)