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

Yes.

The spelling, meaning, and grammatical category (feminine noun) of "esponja" are identical in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazilian Portuguese, the initial "e" is typically pronounced as a clear, open /e/. In Continental Portuguese, unstressed vowels are often reduced or "swallowed," so the "e" is much more closed and sounds closer to a neutral, short /ɨ/.

Brazilian Portuguese Examples

1. Eu usei a esponja para limpar a mesa. (I used the sponge to clean the table.)
2. A esponja de cozinha está muito velha. (The kitchen sponge is very old.)
3. Ele é uma esponja, absorve todo o conhecimento. (He is a sponge, he absorbs all knowledge.)
4. Compre uma esponja macia para o rosto. (Buy a soft sponge for the face.)
5. A esponja de aço estragou a panela. (The steel wool ruined the pan.)

Continental Portuguese Examples

1. Usei a esponja para limpar a mesa. (I used the sponge to clean the table.)
2. A esponja da cozinha está muito velha. (The kitchen sponge is very old.)
3. Ele é uma esponja, absorve todo o conhecimento. (He is a sponge, he absorbs all knowledge.)
4. Compra uma esponja macia para o rosto. (Buy a soft sponge for the face.)
5. A palha de aço estragou a panela. (The steel wool ruined the pan.)