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

Yes

The word "sindicato" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are generally more open, and the final "o" is pronounced clearly as /u/. In Continental Portuguese, there is significant vowel reduction; the unstressed vowels (such as the "i" and the final "o") are much more "closed," shorter, or nearly silent, giving the word a more clipped, rhythmic cadence.

Brazilian Portuguese Examples

1. O sindicato convocou uma assembleia de trabalhadores. (The union called a workers' assembly.)
2. O sindicato está lutando por melhores salários. (The union is fighting for better wages.)
3. Eu sou filiado ao sindicato dos professores. (I am a member of the teachers' union.)
4. O sindicato denunciou o descumprimento das regras. (The union reported the failure to comply with the rules.)
5. A greve foi aprovada pelo sindicato. (The strike was approved by the union.)

Portuguese (Portugal) Examples

1. O sindicato convocou uma assembleia de trabalhadores. (The union called a workers' assembly.)
2. O sindicato está a lutar por melhores salários. (The union is fighting for better wages.)
3. Eu estou filiado ao sindicato dos professores. (I am a member of the teachers' union.)
4. O sindicato denunciou o incumprimento das regras. (The union reported the failure to comply with the rules.)
5. A greve foi aprovada pelo sindicato. (The strike was approved by the union.)