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

Yes

The word "servir" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. There are no differences in its semantic range (it can mean to serve food, to serve a purpose, or to be useful) or its syntax. The only differences lie in pronunciation:

• Brazilian Portuguese: The vowels are generally more open, and the "r" sound varies by region but is often a retroflex "r" (like in American English) or a strong alveolar tap.
• Continental Portuguese: The pronunciation involves more vowel reduction (making the word sound "shorter" or more closed) and the "r" is typically more uvular or a light tap.

Brazilian Portuguese Examples

1. O garçom serviu o jantar muito rápido. (The waiter served dinner very quickly.)
2. Este celular não serve mais para jogar. (This cell phone is no longer useful for playing games.)
3. Para que serve este botão vermelho? (What is this red button for?)
4. O café da manhã está servido! (Breakfast is served!)
5. Esse ônibus não me serve, prefiro o metrô. (That bus doesn't work for me, I prefer the subway.)

Continental Portuguese Examples

1. O empregado serviu o jantar muito rápido. (The waiter served dinner very quickly.)
2. Este telemóvel já não serve para jogar. (This mobile phone is no longer useful for playing games.)
3. Para que é que isto serve? (What is this for?)
4. O pequeno-almoço está servido! (Breakfast is served!)
5. Este autocarro não me serve, prefiro o metro. (This bus doesn't work for me, I prefer the metro.)