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

Yes.

The spelling, meaning, and grammatical function of "início" are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation: Brazilian Portuguese tends to use more open vowels, whereas Continental Portuguese often features more closed or reduced vowel sounds.

Brazilian Portuguese examples:

1. O início do filme foi muito legal. (The start of the movie was very cool.)
2. No início da manhã, estava chovendo muito. (At the beginning of the morning, it was raining a lot.)
3. Precisamos marcar o início da nossa viagem. (We need to schedule the start of our trip.)
4. No início, eu não gostava de estudar matemática. (In the beginning, I didn't like studying math.)
5. O início das aulas vai ser logo ali. (The start of classes will be right there.)

Portuguese (Portugal) examples:

1. O início do filme foi bastante impactante. (The start of the film was quite impactful.)
2. No início da manhã, estava a chover muito. (At the start of the morning, it was raining a lot.)
3. O início do projeto começou-nos a preocupar. (The start of the project began to worry us.)
4. No início da semana, encontramo-nos na biblioteca. (At the beginning of the week, we found ourselves at the library.)
5. O início da sessão está previsto para as dez horas. (The start of the session is scheduled for ten o'clock.)