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

Yes

The word "contrato" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels tend to be more open and the rhythm is more melodic and syllable-timed. In Continental Portuguese, the pronunciation is more "closed"; unstressed vowels (such as the "a" in the middle of the word) are often reduced, shortened, or almost swallowed, making the word sound more consonant-heavy.

Brazilian Portuguese Examples

1. Eu assinei o contrato ontem. (I signed the contract yesterday.)
2. O contrato de aluguel é muito longo. (The rental contract is very long.)
3. Eu contrato novos estagiários todo mês. (I hire new interns every month.)
4. Você leu as cláusulas do contrato? (Did you read the clauses of the contract?)
5. O contrato de trabalho acabou. (The employment contract ended.)

Continental Portuguese Examples

1. Eu assinei o contrato ontem. (I signed the contract yesterday.)
2. O contrato de arrendamento é muito longo. (The lease contract is very long.)
3. Eu contrato novos estagiários todos os meses. (I hire new interns every month.)
4. Já leste as cláusulas do contrato? (Have you read the clauses of the contract?)
5. O contrato de trabalho terminou. (The employment contract ended.)