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

No.

While the spelling, grammar, and fundamental meaning of "realmente" are identical in both Brazilian and Continental Portuguese, there are differences in typical usage and pronunciation.

In terms of usage, although "realmente" is used in both dialects, Brazilians frequently substitute it with "mesmo" or "de verdade" in casual conversation to add emphasis. In Portugal, while "realmente" is perfectly natural, speakers very often use the phrase "de facto" to express the same idea of "indeed" or "in fact." Additionally, the pronunciation differs significantly: Brazilian Portuguese tends to pronounce the vowels more clearly, whereas Continental Portuguese involves heavy vowel reduction, making the first "e" nearly silent and the final "e" sound like a neutralized schwa [ɐ].

Brazilian Portuguese Examples

1. Ele realmente gosta de você. (He really likes you.)
2. Isso é realmente verdade? (Is that really true?)
   3. Realmente, eu não entendi. (Really, I didn't understand.)
3. É realmente muito difícil. (It is really very difficult.)
4. Realmente, você tem razão. (Really, you are right.)

Continental Portuguese Examples

1. Ele realmente gosta de ti. (He really likes you.)
2. Isso é verdade, de facto? (Is that true, indeed?)
3. Realmente, eu não percebi. (Really, I didn't understand.)
4. É mesmo muito difícil. (It is really very difficult.)
5. Realmente, tens razão. (Really, you are right.)