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

Yes

The word "prometi" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Continental Portuguese, unstressed vowels tend to be "reduced" or more closed, whereas in Brazilian Portuguese, vowels are typically more open and clearly articulated.

Brazilian Portuguese Examples

1. Eu prometi para a minha mãe que eu ia chegar cedo. (I promised my mother that I was going to arrive early.)
2. Eu prometi que ia te ligar amanhã. (I promised I was going to call you tomorrow.)
3. Prometi que não ia contar pra ninguém. (I promised I wasn't going to tell anyone.)
4. Eu prometi pro meu amigo que ia ajudar ele. (I promised my friend I was going to help him.)
5. Prometi que ia comprar o doce. (I promised I was going to buy the candy.)

Portuguese (Portugal) Examples

1. Eu prometi à minha mãe que chegaria cedo. (I promised my mother that I would arrive early.)
2. Prometi que te ligaria amanhã. (I promised that I would call you tomorrow.)
3. Prometi que não contaria a ninguém. (I promised that I wouldn't tell anyone.)
4. Prometi ao meu amigo que o ajudaria. (I promised my friend that I would help him.)
5. Prometi que compraria o doce. (I promised that I would buy the candy.)