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

Yes

The differences between "pedra" in Brazilian Portuguese and Continental Portuguese are strictly related to pronunciation and rhythm.

In Brazilian Portuguese, the word is typically pronounced with a more "syllable-timed" rhythm, where the vowels are more open and clearly articulated (e.g., [ˈpɛdɾɐ]). In Continental Portuguese, the rhythm is more "stress-timed," meaning unstressed vowels are often reduced, shortened, or almost swallowed, making the word sound more "closed" or consonant-heavy (e.g., [ˈpɛðɾɐ]). There are no differences in spelling, grammar, or the fundamental meaning of the word.

Brazilian Portuguese Examples

1. Eu tropecei em uma pedra no caminho. (I tripped on a stone on the path.)
2. Essa pedra é muito pesada para eu carregar sozinho. (That stone is very heavy for me to carry alone.)
3. O jardim está cheio de pedras decorativas. (The garden is full of decorative stones.)
4. Ela usa um anel com uma pedra preciosa linda. (She wears a ring with a beautiful gemstone.)
5. Não jogue pedras em quem está tentando ajudar. (Don't throw stones at those trying to help.)

Continental Portuguese Examples

1. Eu tropecei numa pedra no caminho. (I tripped on a stone on the path.)
2. Esta pedra é demasiado pesada para eu carregar sozinho. (This stone is too heavy for me to carry alone.)
3. O jardim está repleto de pedras decorativas. (The garden is full of decorative stones.)
4. Ela usa um anel com uma pedra preciosa lindíssima. (She wears a ring with a very beautiful gemstone.)
5. Não atires pedras a quem está a tentar ajudar. (Don't throw stones at those trying to help.)