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

Yes

The spelling, meaning, and grammatical category of "museu" are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, vowels (especially unstressed ones) tend to be more open and clearly articulated. In Continental Portuguese, unstressed vowels are often more reduced or "swallowed," meaning the final "u" might sound much shorter or more clipped than in Brazil.

Brazilian Portuguese Examples:

1. Eu adoro visitar o museu de arte. (I love visiting the art museum.)
2. O museu está fechado hoje. (The museum is closed today.)
3. Vamos ao museu no próximo domingo? (Shall we go to the museum next Sunday?)
4. Este museu tem uma coleção incrível. (This museum has an incredible collection.)
5. A entrada no museu é gratuita. (Entry to the museum is free.)

Continental Portuguese Examples:

1. Gosto imenso de ir ao museu. (I like going to the museum immensely.)
2. O museu encontra-se encerrado. (The museum is closed.)
3. Fomos ao museu de história ontem. (We went to the history museum yesterday.)
4. Que museu fantástico este! (What a fantastic museum this is!)
5. Pretendo visitar o museu amanhã. (I intend to visit the museum tomorrow.)