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

Yes

The word "ou" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are typically pronounced clearly and openly ([o.u]). In Continental Portuguese, the vowels are often reduced or "swallowed" in natural, rapid speech, making the word sound more closed or even nearly indistinguishable from a single, short [u] sound.

Brazilian Portuguese

1. Você quer café ou chá? (Do you want coffee or tea?)
2. A gente pode ir ao cinema ou ao parque. (We can go to the cinema or to the park.)
3. Você prefere o celular ou o computador? (Do you prefer the cell phone or the computer?)
4. Ele vai de carro ou de ônibus? (Is he going by car or by bus?)
5. Pode ser hoje ou amanhã. (It can be today or tomorrow.)

Continental Portuguese

1. Queres café ou chá? (Do you want coffee or tea?)
2. Podemos ir ao cinema ou ao parque. (We can go to the cinema or to the park.)
3. Preferes o telemóvel ou o computador? (Do you prefer the mobile phone or the computer?)
4. Ele vai de carro ou de autocarro? (Is he going by car or by bus?)
5. Podes ler este livro ou aquele? (Can you read this book or that one?)