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

Yes

The spelling and meaning of the word are identical in both Brazilian and European Portuguese. The only difference is the pronunciation of the final "e." In Brazilian Portuguese, the final "e" is typically pronounced as a clear, unstressed "i" (en-vɛ-lo-pi). In European Portuguese, the final "e" is a "closed" vowel, which is much more muffled and nearly silent (en-vɛ-lɔ-pɨ).

Brazilian Portuguese natural usage:

1. Eu coloquei a carta no envelope. (I put the letter in the envelope.)
2. Você tem um envelope pardo para mim? (Do you have a manila envelope for me?)
3. Preciso comprar envelopes para os convites. (I need to buy envelopes for the invitations.)
4. Escrevi o endereço no envelope. (I wrote the address on the envelope.)
5. O envelope já está lacrado. (The envelope is already sealed.)

European Portuguese natural usage:

1. Coloquei a carta no envelope. (I put the letter in the envelope.)
2. Tens um envelope castanho? (Do you have a brown/manila envelope?)
3. Preciso de comprar envelopes para os convites. (I need to buy envelopes for the invitations.)
4. Escrevi a morada no envelope. (I wrote the address on the envelope.)
5. O envelope está fechado. (The envelope is closed.)