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

Yes

The word "abre" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are generally more open and clearly articulated. In Continental Portuguese, unstressed vowels undergo "vowel reduction," meaning the "e" in "abre" is pronounced much more closed, often sounding like a very short, almost silent [ɨ] or [ə].

Brazilian Portuguese Examples

1. Abre a porta, por favor. (Open the door, please.)
2. Abre o olho com esse cara! (Watch out for this guy!)
3. Abre o jogo e me conta a verdade. (Be honest and tell me the truth.)
4. A loja abre às nove horas. (The store opens at nine o'clock.)
5. Ele abre o presente agora. (He opens the gift now.)

Portuguese (Portugal) Examples

1. Abre a porta, por favor. (Open the door, please.)
2. Fica atento a este tipo! (Watch out for this guy!)
3. Diz-me a verdade e não escondas nada. (Tell me the truth and don't hide anything.)
4. A loja abre às nove horas. (The store opens at nine o'clock.)
5. Ele abre o presente agora. (He opens the gift now.)