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

Yes

The word "regularmente" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference is in pronunciation.

In Brazilian Portuguese, the pronunciation is generally more "open," and the unstressed vowels are clearly articulated. In Continental Portuguese, there is a much stronger tendency toward vowel reduction; the unstressed "e" in the suffix "-mente" is often reduced to a near-silent or very short sound, making the word sound more "clipped" or "closed" to a Brazilian ear.

Brazilian Portuguese Examples

1. Eu vou à academia regularmente. (I go to the gym regularly.)
2. O ônibus passa regularmente por aqui. (The bus passes by here regularly.)
3. Ele usa o celular regularmente para trabalhar. (He uses his cell phone regularly for work.)
4. Nós tomamos café juntos regularmente. (We drink coffee together regularly.)
5. Ela estuda inglês regularmente. (She studies English regularly.)

Continental Portuguese Examples

1. Eu vou ao ginásio regularmente. (I go to the gym regularly.)
2. O autocarro passa regularmente por aqui. (The bus passes by here regularly.)
3. Ele usa o telemóvel regularmente para trabalhar. (He uses his mobile phone regularly for work.)
4. Nós tomamos café juntos regularmente. (We drink coffee together regularly.)
5. Ela estuda inglês regularmente. (She studies English regularly.)