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

Yes

The only differences are in pronunciation. In Brazilian Portuguese, the final "e" is typically pronounced as a short, unstressed "i" (e.g., geralmenti), and the "r" is often a soft tap. In Portugal, the final "e" is much more closed or nearly silent, and the "r" may be articulated with more friction depending on the regional accent. The spelling, meaning, and grammatical usage are identical in both dialects.

Brazilian Portuguese examples:

1. Eu geralmente acordo cedo. (I usually wake up early.)
2. A gente geralmente almoça em casa. (We usually eat lunch at home.)
3. Ele geralmente chega atrasado no trabalho. (He usually arrives late to work.)
4. Você geralmente toma café da manhã o quê? (What do you usually eat for breakfast?)
5. O ônibus geralmente passa por aqui. (The bus usually passes by here.)

European Portuguese examples:

1. Eu geralmente acordo cedo. (I usually wake up early.)
2. Nós geralmente almoçamos em casa. (We usually eat lunch at home.)
3. Ele geralmente chega atrasado ao trabalho. (He usually arrives late to work.)
4. Tu costumas tomar o pequeno-almoço às oito? (Do you usually have breakfast at eight?)
5. O autocarro geralmente passa por aqui. (The bus usually passes by here.)