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

Yes

The word "dente" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the final "e" is typically pronounced as a soft /i/ sound (e.g., den-chee). In Continental Portuguese, the final "e" is much more reduced and often sounds like a near-silent, neutral vowel (e.g., den-t(ɨ)).

Brazilian Portuguese Examples

1. Eu escovei os meus dentes. (I brushed my teeth.)
2. O bebê está nascendo o primeiro dente. (The baby is getting its first tooth.)
3. Eu estou com uma dor de dente muito forte. (I have a very strong toothache.)
4. Ele quebrou um dente jogando futebol. (He broke a tooth playing soccer.)
5. Eu preciso ir ao dentista para cuidar dos dentes. (I need to go to the dentist to take care of my teeth.)

Continental Portuguese Examples

1. Escovei os dentes. (I brushed my teeth.)
2. O bebé está a nascer o primeiro dente. (The baby is getting its first tooth.)
3. Estou com uma dor de dente muito forte. (I have a very strong toothache.)
4. Ele partiu um dente a jogar futebol. (He broke a tooth playing soccer.)
5. Preciso de ir ao dentista para tratar os dentes. (I need to go to the dentist to treat my teeth.)