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

No.

While the fundamental meaning of "provador" (a place to try on clothes) is shared by both varieties, there is a difference in typical usage. In Brazil, "provador" is the standard and most natural term for a clothing fitting room. In Portugal, while "provador" is understood, it is much more common and natural for people to use the terms "cabine" or "cabine de prova."

Brazilian Portuguese Examples

1. Onde fica o provador? (Where is the fitting room?)
2. O provador está ocupado. (The fitting room is occupied.)
3. Tem muita fila no provador. (There is a long line at the fitting room.)
4. Vou levar essa blusa para o provador. (I'm going to take this blouse to the fitting room.)
5. O provador da loja é bem espaçoso. (The store's fitting room is quite spacious.)

Portuguese (Continental) Examples

1. Onde é a cabine? (Where is the fitting room?)
2. A cabine está ocupada. (The fitting room is occupied.)
3. Há muita fila para a cabine de prova. (There is a long line for the fitting room.)
4. Vou experimentar esta camisola na cabine. (I'm going to try this sweater on in the fitting room.)
5. A cabine de prova é muito pequena. (The fitting room is very small.)