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

No

The difference between the two is not in the spelling, grammar, or meaning of the word itself, but in its typical usage. In Brazil, the verb "beber" is most commonly associated with consuming alcoholic beverages. When referring to non-alcoholic liquids like water, juice, or soda, Brazilians much more naturally use the verb tomar. In Portugal, "beber" is the standard, universal verb used for all types of liquids, whether they contain alcohol or not.

Brazilian Portuguese

1. Ele bebe muito quando sai com os amigos. (He drinks a lot when he goes out with friends.)
2. Vamos beber uma cerveja gelada? (Shall we drink a cold beer?)
3. Você gosta de beber cachaça? (Do you like to drink cachaça?)
4. Ele parou de beber para cuidar da saúde. (He stopped drinking to take care of his health.)
5. Quer beber algo agora? (Do you want to drink something now?)

Portuguese (Portugal)

1. Ele bebe muito quando sai com os amigos. (He drinks a lot when he goes out with friends.)
2. Vamos beber uma cerveja gelada? (Shall we drink a cold beer?)
3. Gostas de beber aguardente? (Do you like to drink aguardente?)
4. Ele parou de beber para cuidar da saúde. (He stopped drinking to take care of his health.)
5. Queres beber alguma coisa agora? (Do you want to drink something now?)