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

Yes.

The only difference between the two variants is pronunciation. In Brazilian Portuguese, unstressed vowels—specifically the "e" in gaveta—tend to be more open and clearly articulated. In Continental Portuguese, unstressed vowels undergo significant reduction, meaning the "e" is much more closed and can sound almost like a "schwa" or even disappear entirely, leading to a more "clipped" sound.

Brazilian Portuguese Examples

1. "A chave está na gaveta da cozinha." (The key is in the kitchen drawer.)
2. "Eu perdi o meu carregador em alguma gaveta." (I lost my charger in some drawer.)
3. "Essa gaveta está muito difícil de abrir." (This drawer is very difficult to open.)
4. "Coloque os documentos na gaveta de baixo." (Put the documents in the bottom drawer.)
5. "Limpei todas as gavetas do meu quarto hoje." (I cleaned all the drawers in my room today.)

Portuguese (Portugal) Examples

1. "Onde está a chave? Esqueci-a na gaveta." (Where is the key? I forgot it in the drawer.)
2. "Esta gaveta está cheia de tralha." (This drawer is full of junk.)
3. "Põe as canetas na gaveta, por favor." (Put the pens in the drawer, please.)
4. "A gaveta está sempre encravada." (The drawer is always stuck.)
5. "Vou arrumar a gaveta das meias." (I'm going to tidy the sock drawer.)