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

Yes

The word "dúvida" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, vowels in unstressed syllables are generally more open and clearly articulated, so the "i" in dúvida is clearly pronounced. In Continental Portuguese, unstressed vowels tend to be more closed or even reduced/elided, making the word sound much more compressed, as if the "i" were nearly silent.

Brazilian Portuguese Examples

1. Eu tenho uma dúvida sobre a matéria. (I have a doubt/question about the subject.)
2. Não fique na dúvida, pode perguntar. (Don't stay in doubt, you can ask.)
3. Fiquei em dúvida sobre qual caminho seguir. (I was in doubt about which path to follow.)
4. Sem dúvida nenhuma, ele é o melhor. (Without any doubt, he is the best.)
5. Minha única dúvida é sobre o horário. (My only doubt is about the time.)

Continental Portuguese Examples

1. Tenho uma dúvida relativamente à matéria. (I have a doubt regarding the subject.)
2. Não fiques com dúvidas, pergunta ao professor. (Don't be in doubt, ask the teacher.)
3. Fiquei com uma dúvida sobre qual caminho seguir. (I was left with a doubt about which path to follow.)
4. Sem dúvida alguma, ele é o melhor. (Without any doubt at all, he is the best.)
5. A minha dúvida é quanto ao horário. (My doubt is regarding the time.)