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

Yes

The meaning, spelling, and grammar of the word "fatura" are identical in both Brazilian and European Portuguese. The difference lies entirely in pronunciation.

In Brazilian Portuguese, the vowels are more "open" and the rhythm is more melodic (syllable-timed). The final "a" is clearly articulated. In Continental (European) Portuguese, unstressed vowels are much more "closed" or reduced. An inhabitant of Lisbon might pronounce the final "a" in "fatura" so subtly that it almost disappears, making the word sound more clipped and consonant-heavy.

Brazilian Portuguese Examples

1. Eu esqueci de pagar a fatura do cartão. (I forgot to pay the credit card bill.)
2. A fatura da internet veio mais cara este mês. (The internet bill came more expensive this month.)
3. Você pode conferir a fatura para mim? (Can you check the invoice for me?)
4. A fatura já foi paga pelo meu pai. (The bill has already been paid by my father.)
5. Tem um erro nesta fatura aqui. (There is an error in this invoice here.)

Portuguese (Portugal) Examples

1. Já recebi a fatura do cartão. (I have already received the credit card bill.)
2. É necessário pagar a fatura da eletricidade. (It is necessary to pay the electricity bill.)
3. Poderia enviar-me a fatura, por favor? (Could you send me the invoice, please?)
4. O valor da fatura não coincide com o pedido. (The invoice amount does not match the order.)
5. Verifiquei a fatura e está tudo correto. (I checked the invoice and everything is correct.)