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

Yes

The only differences between the word "obrigado" in Brazilian and Continental Portuguese are in pronunciation and phonology.

• Pronunciation: In Brazilian Portuguese, the vowels are generally more "open" and the rhythm is syllable-timed, meaning the "o" in obrigado is clearly articulated. In Continental Portuguese, there is significant vowel reduction; unstressed vowels are often "swallowed" or shortened, which can make the word sound more closed or even clipped to a Brazilian ear.
• Phonetic variation: Depending on the region of Brazil (such as Rio de Janeiro or São Paulo), the "d" might be pronounced with a slight "dj" sound (palatalization), whereas in Portugal, the "d" remains dental.

Brazilian Portuguese Examples

1. Muito obrigado pela ajuda! (Thank you very much for the help!)
2. Valeu, cara! Muito obrigado. (Thanks, man! Thank you very much.)
3. Obrigado pelo presente, viu? (Thank you for the gift, okay?)
4. Muito obrigado por tudo, pessoal. (Thank you very much for everything, everyone.)
5. Obrigado, moço. (Thank you, young man.)

Continental Portuguese Examples

1. Muito obrigado pela ajuda. (Thank you very much for the help.)
2. Obrigado, pá! (Thanks, man/mate!)
3. Muito obrigado por tudo. (Thank you very much for everything.)
4. Fico-lhe muito agradecido. (I am very grateful to you.)
5. Muito obrigado, senhor. (Thank you very much, sir.)