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

Yes

The word "dinheiro" is identical in spelling, meaning, and grammatical function in both Brazilian and European Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are generally more open, and the nasalization of the "in" syllable is very prominent. In European Portuguese, the pronunciation tends to be more closed, and the vowels are often reduced or "swallowed" in unstressed positions, resulting in a more staccato rhythm.

Brazilian Portuguese examples:

1. Eu não tenho dinheiro para comprar esse carro. (I don't have money to buy that car.)
2. Quanto dinheiro você gastou no shopping? (How much money did you spend at the mall?)
3. É preciso economizar dinheiro para o futuro. (It is necessary to save money for the future.)
4. Ele sempre tem dinheiro para o lanche. (He always has money for a snack.)
5. O dinheiro sumiu da minha carteira. (The money disappeared from my wallet.)

European Portuguese examples:

1. Não tenho dinheiro para comprar este carro. (I don't have money to buy this car.)
2. Quanto dinheiro gastaste no centro comercial? (How much money did you spend at the shopping mall?)
3. É preciso poupar dinheiro para o futuro. (It is necessary to save money for the future.)
4. Ele tem sempre dinheiro para o lanche. (He always has money for a snack.)
5. O dinheiro desapareceu da minha carteira. (The money disappeared from my wallet.)