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

Yes

The word "da" is a contraction of the preposition de (of/from) and the feminine definite article a (the). There are no differences in its meaning, grammatical function, or spelling between Brazilian and Continental Portuguese. The only differences are in pronunciation: Brazilian Portuguese tends to pronounce the vowels in the contraction more openly, while Continental Portuguese often features more reduced or closed vowel sounds in unstressed syllables.

Brazilian Portuguese

1. A porta da casa está aberta. (The door of the house is open.)
2. Eu gosto da música brasileira. (I like Brazilian music.)
3. Ela é a melhor da turma. (She is the best in the class.)
4. O motorista do ônibus é legal. (The bus driver is nice.)
5. A bateria do celular acabou. (The cell phone battery ran out.)

Continental Portuguese

1. A porta da casa está aberta. (The door of the house is open.)
2. Eu gosto da música brasileira. (I like Brazilian music.)
3. Ela é a melhor da turma. (She is the best in the class.)
4. O motorista do autocarro é simpático. (The bus driver is nice.)
5. A bateria do telemóvel acabou. (The cell phone battery ran out.)