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

No

While the spelling and core definition of the word are identical, there are significant differences in pronunciation and typical usage:

• Pronunciation: In Brazilian Portuguese, the "e" is typically more open and clearly articulated (pro-fe-SSOR). In Continental Portuguese, the "e" is often reduced or nearly silent, making the word sound much more closed and compressed (pr(ə)-fe-SSOR).
• Usage: In Brazil, "professor" is used very broadly to describe anyone teaching a skill, including gym instructors, music tutors, or driving instructors. In Portugal, the term is more strictly reserved for academic or school-related contexts. For other types of instruction, a Portuguese person would more naturally use words like instrutor (instructor) or treinador (trainer/coach).

Brazilian Portuguese Examples

1. O professor de matemática é muito bom. (The math teacher is very good.)
2. Minha professora de inglês chegou. (My English teacher arrived.)
3. Ele é professor de musculação na minha academia. (He is a gym instructor at my gym.)
4. Preciso falar com o professor de violão. (I need to speak with the guitar teacher.)
5. O professor de artes explicou o quadro. (The art teacher explained the painting.)

Continental Portuguese Examples

1. O professor de matemática é muito bom. (The math teacher is very good.)
2. A minha professora de inglês chegou. (My English teacher arrived.)
3. O instrutor de musculação da minha academia deu um novo treino. (The gym instructor at my gym gave a new workout.)
4. Preciso de falar com o professor de música. (I need to speak with the music teacher.)
5. O professor de artes explicou o quadro. (The art teacher explained the painting.)