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

Yes

The word "cerveja" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazil, the vowels are typically more open and clearly articulated. In Portugal, unstressed vowels (such as the initial "e") are often "reduced" or more closed, making the first syllable sound much shorter or more subtle.

Brazilian Portuguese Examples

1. Eu quero uma cerveja bem gelada, por favor. (I want a very cold beer, please.)
2. Vamos tomar uma cerveja depois do trabalho? (Shall we grab a beer after work?)
3. Essa cerveja artesanal é muito boa. (This craft beer is very good.)
4. Não gosto de cerveja sem gás. (I don't like flat beer.)
5. A cerveja acabou, preciso comprar mais. (The beer ran out, I need to buy more.)

Portuguese (Continental) Examples

1. Queria uma imperial, por favor. (I would like a draught beer, please.)
2. Vamos beber uma cerveja ao fim da tarde? (Let's have a beer in the late afternoon?)
3. Esta cerveja está muito fresca. (This beer is very fresh.)
4. Gostas de beber cerveja ou vinho? (Do you like to drink beer or wine?)
5. Podes trazer mais uma cerveja? (Can you bring another beer?)