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

Yes

The word is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The difference is strictly phonetic: in Brazil, the "e" is typically an open vowel ([ɛ]) and the "s" is a standard [s] sound. In Portugal, the "s" at the end of the word is typically pronounced as a "sh" sound ([ʃ]), and the vowel may be more closed.

Brazilian Portuguese

1. Este mês está passando muito rápido. (This month is passing very quickly.)
2. Eu vou viajar no próximo mês. (I am going to travel next month.)
3. O pagamento será feito no início do mês. (The payment will be made at the beginning of the month.)
4. Quantos meses faltam para o Natal? (How many months are left until Christmas?)
5. Todo mês eu vou à academia. (Every month I go to the gym.)

Continental Portuguese

1. Este mês está a passar muito depressa. (This month is passing very quickly.)
2. Vou viajar no próximo mês. (I am going to travel next month.)
3. O pagamento será feito no início do mês. (The payment will be made at the beginning of the month.)
4. Quantos meses faltam para o Natal? (How many months are left until Christmas?)
5. Todos os meses vou ao ginásio. (Every month I go to the gym.)