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

Yes

The meaning, spelling, and grammatical function of "semestre" are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation.

In Brazilian Portuguese, the vowels are generally more open, and the unstressed "e" at the end of the word is clearly pronounced as a light "ee" sound (like in "see"). In Continental Portuguese, unstressed vowels are often reduced or "swallowed"; the final "e" is much more closed and may sound almost like a silent or very faint "uh" sound, making the word sound more clipped.

Brazilian Portuguese Examples

1. Eu terminei o primeiro semestre da faculdade. (I finished the first semester of college.)
2. As notas do semestre saem na próxima semana. (The semester grades come out next week.)
3. O segundo semestre vai ser muito puxado. (The second semester is going to be very tough.)
4. Precisamos revisar todo o conteúdo do semestre. (We need to review all the content from the semester.)
5. Ele passou em todas as matérias do semestre. (He passed all the subjects of the semester.)

Continental Portuguese Examples

1. Concluí o primeiro semestre da licenciatura. (I completed the first semester of my degree.)
2. Os resultados do semestre serão publicados brevemente. (The semester results will be published shortly.)
3. O próximo semestre letivo começa em setembro. (The next academic semester begins in September.)
4. Tive muito trabalho durante o último semestre. (I had a lot of work during the last semester.)
5. As inscrições para o semestre estão abertas. (Enrollments for the semester are open.)