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

Yes

The spelling, meaning, and grammatical function of "responsável" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the vowels are typically more open, and the final "l" is pronounced as a "u" sound (e.g., res-pon-sá-veu). In Continental Portuguese, the vowels are often more closed or reduced, and the "l" is a clear, dentalized consonant (e.g., res-pon-sá-vel).

Brazilian Portuguese Examples

1. Ele é o responsável pelo projeto. (He is responsible for the project.)
2. Ela é uma pessoa muito responsável. (She is a very responsible person.)
3. O responsável deve assinar o documento. (The person in charge must sign the document.)
4. Quem é o responsável por essa confusão? (Who is responsible for this mess?)
5. Precisamos de um uso responsável da água. (We need a responsible use of water.)

Continental Portuguese Examples

1. O responsável pela obra já saiu. (The person in charge of the construction has already left.)
2. Tu tens de ser mais responsável com os teus estudos. (You have to be more responsible with your studies.)
3. Os pais ou responsáveis devem vir à reunião. (Parents or guardians must come to the meeting.)
4. Não sei quem é o responsável por este incidente. (I don't know who is responsible for this incident.)
5. É fundamental ter uma atitude responsável. (It is fundamental to have a responsible attitude.)