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

Yes

The word "finalmente" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. It is used to express that something has happened after a long period of waiting, or to denote the final step in a sequence. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels tend to be more open and clearly articulated. In Continental Portuguese, there is a much stronger tendency toward vowel reduction, meaning the unstressed vowels (especially the "e" at the end) may become almost silent or sound like a very short "schwa" sound.

Brazilian Portuguese Examples

1. Finalmente, as férias chegaram! (Finally, the holidays have arrived!)
2. Finalmente consegui achar meu celular. (I finally managed to find my cell phone.)
3. O ônibus finalmente passou. (The bus finally passed/arrived.)
4. Finalmente, você chegou! (Finally, you've arrived!)
5. A comida finalmente ficou pronta. (The food is finally ready.)

Portuguese (Continental) Examples

1. Finalmente, as férias começaram! (Finally, the holidays have started!)
2. Finalmente, encontrei o meu telemóvel. (Finally, I found my mobile phone.)
3. O comboio finalmente passou. (The train finally passed/arrived.)
4. Finalmente, tu chegaste! (Finally, you arrived!)
5. A comida finalmente está pronta. (The food is finally ready.)