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

Yes

The only difference between "parabéns" in Brazilian and Continental Portuguese is pronunciation. In Portugal, the final "s" is typically pronounced as a "sh" sound ([paɾɐˈbɛ̃ʃ]), whereas in most regions of Brazil, it is pronounced as an "s" sound ([paɾɐˈbɛ̃s]). The meaning, spelling, and grammatical usage are identical in both dialects.

Brazilian Portuguese Examples

1. Parabéns pelo seu aniversário! (Happy birthday!)
2. Muitíssimo parabéns pela conquista! (Huge congratulations on the achievement!)
3. Passando para te dar os parabéns! (Just stopping by to congratulate you!)
4. Parabéns, você merece muito! (Congrats, you really deserve it!)
5. Meus parabéns pelo novo emprego! (My congratulations on the new job!)

Continental Portuguese Examples

1. Muitos parabéns pelo teu aniversário! (Many congratulations on your birthday!)
2. Parabéns pelo teu sucesso! (Congrats on your success!)
3. Dou-te os parabéns pela nota! (I congratulate you on the grade!)
4. Muitos parabéns pela tua vitória! (Many congratulations on your victory!)
5. Parabéns por teres conseguido! (Congrats on having managed to do it!)