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

Yes

The word "bolso" is identical in meaning, spelling, and grammatical gender (masculine) in both Brazilian and Continental Portuguese. It refers specifically to the pocket of a piece of clothing. The only difference lies in pronunciation: in Continental Portuguese, the unstressed vowels are more likely to be reduced or "swallowed" (a phenomenon called vowel reduction), whereas in Brazilian Portuguese, the vowels are typically pronounced more clearly and openly. While the word "bolso" remains the same, the surrounding vocabulary in a natural sentence (such as the names for clothing or objects) often differs between the two regions.

Brazilian Portuguese Examples

1. Esqueci meu celular no bolso da calça. (I forgot my cell phone in my pants pocket.)
2. Ele colocou a carteira no bolso da jaqueta. (He put his wallet in the jacket pocket.)
3. Tem um papel no bolso da sua bermuda. (There is a piece of paper in your shorts pocket.)
4. Meu bolso está cheio de moedas. (My pocket is full of coins.)
5. Não deixe nada no bolso da calça jeans. (Don't leave anything in your jeans pocket.)

Continental Portuguese Examples

1. Esqueci o telemóvel no bolso das calças. (I forgot my mobile phone in my trousers pocket.)
2. Ele pôs a carteira no bolso do casaco. (He put his wallet in the coat pocket.)
3. Há um papel no bolso da tua bermuda. (There is a piece of paper in your shorts pocket.)
4. O meu bolso está cheio de moedas. (My pocket is full of coins.)
5. Não deixes nada no bolso das calças de ganga. (Don't leave anything in your denim trousers pocket.)