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

No.

While the meaning, spelling, and fundamental grammar of "que" are identical in both dialects, there are significant differences in typical usage and pronunciation.

In terms of usage, Brazilian Portuguese (BP) often employs the colloquial construction "que que" in informal questions (a reduction of "o que que"), whereas Continental Portuguese (EP) typically uses the emphatic particle "é que" (e.g., "O que é que..."). Furthermore, the pronunciation differs: in Brazil, the "e" in "que" is usually more open or clearly articulated, while in Portugal, the vowel is frequently subject to heavy reduction, often becoming almost silent in unstressed positions.

Brazilian Portuguese Examples

1. Que que você quer comer? (What do you want to eat?)
2. O cachorro que eu vi é muito lindo. (The dog that I saw is very beautiful.)
3. O que você está fazendo aí? (What are you doing there?)
4. Espero que você chegue logo. (I hope that you arrive soon.)
5. O que você quer de mim? (What do you want from me?)

Continental Portuguese Examples

1. O que é que tu queres comer? (What is it that you want to eat?)
2. O cão que eu vi é muito lindo. (The dog that I saw is very beautiful.)
3. O que é que estás a fazer aí? (What is it that you are doing there?)
4. Espero que venhas logo. (I hope that you come soon.)
5. O que é que tu queres de mim? (What is it that you want from me?)