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

Yes

The word "duvidar" is identical in both Brazilian and Continental Portuguese regarding its meaning, spelling, and grammatical function. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are typically more open and clearly articulated. In Continental Portuguese, unstressed vowels are often more closed or partially elided (devoiced), resulting in a more consonant-heavy sound.

Brazilian Portuguese Examples

1. Eu não duvido que ele venha à festa. (I don't doubt that he will come to the party.)
2. Ela duvida de tudo o que eu falo. (She doubts everything I say.)
3. Não duvide do seu próprio talento. (Don't doubt your own talent.)
4. Todo mundo duvida dessa história estranha. (Everyone doubts this strange story.)
5. Eu duvido muito que o ônibus chegue no horário. (I highly doubt that the bus will arrive on time.)

Continental Portuguese Examples

1. Não duvido que ele venha à festa. (I don't doubt that he will come to the party.)
2. Ela duvida de tudo o que eu digo. (She doubts everything I say.)
3. Não duvides do teu próprio talento. (Don't doubt your own talent.)
4. Toda a gente duvida desta história estranha. (Everyone doubts this strange story.)
5. Duvido bastante que o autocarro chegue à hora prevista. (I doubt quite a bit that the bus will arrive at the scheduled time.)