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

Yes.

The word "Pedro" is a proper noun (a given name), so its meaning, spelling, and grammatical function are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. Brazilian Portuguese tends to have more open vowels and a more rhythmic, syllable-timed cadence. In Continental Portuguese, vowels (especially unstressed ones) are often more closed or reduced, and the rhythm is more stress-timed, giving the name a more "clipped" sound.

Brazilian Portuguese

1. O Pedro é meu melhor amigo. (Pedro is my best friend.)
2. O Pedro vai ao cinema amanhã. (Pedro is going to the movies tomorrow.)
3. Você viu o Pedro na festa? (Did you see Pedro at the party?)
4. Eu preciso falar com o Pedro agora. (I need to talk to Pedro now.)
5. O Pedro trabalha numa padaria. (Pedro works in a bakery.)

Continental Portuguese

1. O Pedro é o meu melhor amigo. (Pedro is my best friend.)
2. O Pedro vai ao cinema amanhã. (Pedro is going to the cinema tomorrow.)
3. Já viste o Pedro na festa? (Have you seen Pedro at the party?)
4. Preciso de falar com o Pedro agora. (I need to talk to Pedro now.)
5. O Pedro trabalha numa pastelaria. (Pedro works in a pastry shop.)