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

Yes

The meaning, spelling, and grammatical usage of "fim" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the final "m" results in a clearly nasalized vowel that is often more open and resonant. In Continental Portuguese, the "m" is much more subtle, serving primarily to nasalize the preceding vowel, which tends to be more closed or reduced in sound.

Brazilian Portuguese Examples

1. "O fim do namoro foi muito triste." (The end of the relationship was very sad.)
2. "Não vejo o fim desse filme." (I don't see the end of that movie.)
3. "O fim de semana vai ser muito legal." (The weekend is going to be very cool.)
4. "O fim da tarde é o meu momento favorito." (The end of the afternoon is my favorite moment.)
5. "Estamos chegando ao fim do projeto." (We are arriving at the end of the project.)

Continental Portuguese Examples

1. "O fim do namoro foi muito triste." (The end of the relationship was very sad.)
2. "Não vejo o fim deste filme." (I don't see the end of this movie.)
3. "O fim de semana vai ser muito fixe." (The weekend is going to be very cool.)
4. "O fim da tarde é o meu momento preferido." (The end of the afternoon is my preferred moment.)
5. "Estamos a chegar ao fim do projeto." (We are arriving at the end of the project.)