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

No.

While the spelling, meaning, and conjugation of "poder" are identical in both varieties, there are differences in pronunciation and typical usage:

• Pronunciation: Brazilian Portuguese is more vocalic and features more open, melodic vowels. In Continental Portuguese, unstressed vowels are often reduced or "swallowed," making the word sound more closed or shorter.
• Typical Usage: In Brazil, "poder" is frequently used to express the successful completion of an action (acting as a synonym for conseguir). In Portugal, "conseguir" is strictly used for "managing to do something," while "poder" is reserved for permission or possibility.

Brazilian Portuguese

1. Eu não posso ir ao shopping hoje. (I can't go to the mall today.)
2. Você pode me ajudar com as sacolas? (Can you help me with the bags?)
3. Eu pude terminar a tarefa a tempo. (I was able to finish the task on time.)
4. Você pode falar um pouco mais alto? (Can you speak a little louder?)
5. A gente pode marcar um lanche mais tarde? (Can we arrange a snack later?)

Continental Portuguese

1. Eu não posso ir ao centro comercial hoje. (I can't go to the shopping center today.)
2. Podes ajudar-me com os sacos? (Can you help me with the bags?)
3. Eu consegui terminar a tarefa a tempo. (I managed to finish the task on time.)
4. Pode falar um pouco mais alto, por favor? (Can you speak a little louder, please?)
5. Podemos combinar um lanche mais tarde? (Can we arrange a snack later?)