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

Yes

The meaning, spelling, and grammatical function of "advogado" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are generally more open and the rhythm is more "syllable-timed" (each syllable receives roughly equal weight). In Continental Portuguese, vowels—especially unstressed ones—are much more closed or nearly silent, and the rhythm is "stress-timed," which can make the word sound shorter and more clipped to a Brazilian ear.

Brazilian Portuguese Examples

1. Eu preciso falar com um advogado. (I need to speak with a lawyer.)
2. O advogado está analisando o meu contrato. (The lawyer is analyzing my contract.)
3. Ela é uma advogada muito competente. (She is a very competent lawyer.)
4. Você já chamou o seu advogado? (Have you already called your lawyer?)
5. Ele decidiu se tornar advogado. (He decided to become a lawyer.)

Continental Portuguese Examples

1. Eu preciso de falar com um advogado. (I need to speak with a lawyer.)
2. O advogado está a analisar o meu contrato. (The lawyer is analyzing my contract.)
3. Ela é uma advogada muito competente. (She is a very competent lawyer.)
4. Já chamaste o teu advogado? (Have you already called your lawyer?)
5. Ele decidiu tornar-se advogado. (He decided to become a lawyer.)