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

Yes

The word "bengala" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. It refers to a walking stick or cane. The only difference is pronunciation. In Brazilian Portuguese, the vowels are typically more open and the rhythm is more syllable-timed. In Continental Portuguese, vowels (especially unstressed ones) tend to be more closed or even reduced/elided, and the rhythm is more stress-timed.

Brazilian Portuguese

1. O meu avô usa uma bengala para caminhar. (My grandfather uses a cane to walk.)
2. Comprei uma bengala nova de madeira. (I bought a new wooden cane.)
3. Ele apoiou a bengala na parede. (He leaned the cane against the wall.)
4. A bengala dele é muito elegante. (His cane is very elegant.)
5. Ela precisa de uma bengala para subir as escadas. (She needs a cane to climb the stairs.)

Continental Portuguese

1. O meu avô usa uma bengala para andar. (My grandfather uses a cane to walk.)
2. Comprei uma bengala nova de madeira. (I bought a new wooden cane.)
3. Ele encostou a bengala na parede. (He leaned the cane against the wall.)
4. A bengala dele é muito elegante. (His cane is very elegant.)
5. Ela precisa de uma bengala para subir as escadas. (She needs a cane to climb the stairs.)