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

Yes.

The word abrir is identical in meaning, spelling, and grammatical classification (infinitive verb) in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazil, the final "r" is often pronounced as an aspirated "h" (in Rio de Janeiro) or a retroflex "r" (in the interior), and the vowels are generally more open. In Portugal, the "r" is typically a more subtle alveolar tap, and the vowels are often more closed or reduced.

Brazilian Portuguese Examples

• Você pode abrir a janela, por favor? (Can you open the window, please?)
• Eu vou abrir uma conta no banco hoje. (I am going to open a bank account today.)
• A loja está abrindo agora. (The shop is opening now.)
• Ele esqueceu de abrir o presente. (He forgot to open the gift.)
• Eu preciso abrir esta garrafa de vinho. (I need to open this bottle of wine.)

Continental Portuguese Examples

• Podes abrir a janela, por favor? (Can you open the window, please?)
• Vou abrir uma conta no banco hoje. (I am going to open a bank account today.)
• A loja está a abrir agora. (The shop is opening now.)
• Ele esqueceu-se de abrir o presente. (He forgot to open the gift.)
• Eu preciso de abrir esta garrafa de vinho. (I need to open this bottle of wine.)