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

Yes

The word "pode" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the final "e" is typically pronounced as a clear, bright /i/ sound (sounding like "poh-jee" in many regions). In Continental Portuguese, unstressed vowels are much more reduced; the final "e" is often a very short, closed /ɨ/ sound, or it may be almost entirely silent, making the word sound more like "pôd" to an untrained ear.

Brazilian Portuguese Examples

1. Você pode me ajudar com a lição? (Can you help me with the homework?)
2. Ele pode chegar um pouco atrasado hoje. (He might arrive a little late today.)
3. Isso não pode acontecer de novo! (This cannot happen again!)
4. Você pode entrar, por favor? (You may enter, please?)
5. Pode deixar, eu resolvo esse problema. (Don't worry, I'll handle this problem.)

Continental Portuguese Examples

1. Pode ser que ele venha à festa. (It might be that he comes to the party.)
2. Não se pode fumar neste recinto. (One cannot smoke in this area.)
3. Ele pode estar a trabalhar agora. (He might be working right now.)
4. Pode falar mais devagar, por favor? (Can [you] speak more slowly, please?)
5. Isto pode ser um grande erro. (This can be a big mistake.)