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

Yes

The word is identical in terms of meaning, spelling, and grammar. It functions as an adverb meaning "fortunately" or "luckily" in both dialects. The only difference is pronunciation. In Brazilian Portuguese, the unstressed vowels are typically more open and clearly articulated. In European Portuguese, unstressed vowels undergo significant reduction, meaning they are shorter or more "swallowed," which alters the rhythmic cadence of the word.

Brazilian Portuguese Examples

1. Felizmente, o ônibus chegou na hora. (Fortunately, the bus arrived on time.)
2. Felizmente, eu achei meu celular. (Luckily, I found my cell phone.)
3. Felizmente, a galera se divertiu muito. (Fortunately, the group/folks had a lot of fun.)
4. Felizmente, o trem não quebrou. (Luckily, the train didn't break down.)
5. Felizmente, o lanche estava uma delícia. (Fortunately, the snack was delicious.)

European Portuguese Examples

1. Felizmente, o autocarro chegou na hora. (Fortunately, the bus arrived on time.)
2. Felizmente, eu encontrei o meu telemóvel. (Luckily, I found my mobile phone.)
3. Felizmente, a malta se divertiu muito. (Fortunately, the gang/group had a lot of fun.)
4. Felizmente, o comboio não avariou. (Luckily, the train didn't break down.)
5. Felizmente, o lanche estava uma delícia. (Fortunately, the snack was delicious.)