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

No.

While the fundamental meaning of "banho" (bath or wash) is the same in both dialects, there is a significant difference in typical usage regarding daily hygiene. In Brazil, "banho" is the standard, ubiquitous word used for both a bath in a tub and a daily shower. In Portugal, "duche" is the much more common and natural word for a daily shower; "banho" in Portugal is more specifically associated with a bath in a tub or the general act of washing an object or an animal.

Brazilian Portuguese Examples

1. Eu vou tomar um banho agora. (I'm going to take a shower now.)
2. O cachorro precisa de um banho. (The dog needs a bath.)
3. O banho estava muito gelado! (The shower was very cold!)
4. Ele gosta de tomar banho de chuva. (He likes to play in the rain.)
5. Vou dar um banho no bebê. (I'm going to give the baby a bath.)

Portuguese (Portugal) Examples

1. Eu vou tomar um duche agora. (I'm going to take a shower now.)
2. O cão precisa de um banho. (The dog needs a bath.)
3. O duche estava muito frio! (The shower was very cold!)
4. Ele gosta de apanhar chuva. (He likes to get caught in the rain.)
5. Vou dar um banho ao bebé. (I'm going to give the baby a bath.)