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

Yes

The word "obrigada" is identical in meaning, grammar, spelling, and fundamental usage in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are generally more "open" and clearly articulated. In Continental Portuguese, vowels—especially those in unstressed syllables—are often "closed" or reduced, meaning the final "a" may sound much shorter or more neutralized.

Brazilian Portuguese Examples

1. Muito obrigada pela ajuda! (Thank you very much for the help!)
2. Valeu, muito obrigada! (Thanks, thank you very much!)
3. Obrigada, viu? (Thanks, okay?)
4. Muito obrigada pelo presente, adorei! (Thank you very much for the gift, I loved it!)
5. Não precisa não, muito obrigada. (No need, thank you very much.)

Portuguese (Continental) Examples

1. Muito obrigada, é muito giro! (Thank you very much, it is very nice!)
2. Muito obrigada pela comida. (Thank you very much for the food.)
3. Obrigada, o autocarro está a chegar. (Thank you, the bus is arriving.)
4. Muito obrigada por me teres ajudado. (Thank you very much for having helped me.)
5. Muito obrigada, estavas impecável. (Thank you very much, you were great.)