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

Yes

The word "faço" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, vowels are generally more open and the rhythm is more syllable-timed (each syllable has a similar duration). In Continental Portuguese, unstressed vowels are often more reduced or closed, and the rhythm is more stress-timed (the rhythm is determined by the beats of the stressed syllables).

Brazilian Portuguese

1. Eu faço academia todos os dias. (I go to the gym every day.)
2. Eu faço o meu dever de América. (I do my homework.) — Wait, let me correct that to a more natural example: Eu faço o meu dever de casa. (I do my homework.)
3. Eu faço um curso de fotografia. (I am taking a photography course.)
4. Eu faço uma viagem de ônibus. (I am taking a trip by bus.)
5. Eu faço o seu bolo. (I am making your cake.)

Continental Portuguese

1. Eu faço exercício todos os dias. (I do exercise every day.)
2. Eu faço o meu trabalho de casa. (I do my homework.)
3. Eu faço um curso de fotografia. (I am taking a photography course.)
4. Eu faço uma viagem de autocarro. (I am taking a trip by bus.)
5. Eu faço o teu bolo. (I am making your cake.)