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

Yes

The word "sogro" is identical in both Brazilian and Continental Portuguese in terms of meaning (father-in-law), spelling, and grammar. The only difference is in pronunciation. In Brazil, the vowels tend to be more open, and the "o" at the end of the word is pronounced clearly as [u]. In Portugal, the pronunciation is more "closed" or reduced, and the rhythm of the word follows the stress-timed nature of European Portuguese, which can make the unstressed syllables sound much shorter.

Brazilian Portuguese Examples

1. Meu sogro é muito legal. (My father-in-law is very cool.)
2. Eu vou almoçar na casa do meu sogro. (I am going to have lunch at my father-in-law's house.)
3. Eu me dou muito bem com o meu sogro. (I get along very well with my father-in-law.)
4. O meu sogro me deu um relógio de presente. (My father-in-law gave me a watch as a gift.)
5. Você conhece o meu sogro? (Do you know my father-in-law?)

Portuguese (Continental) Examples

1. O meu sogro é muito porreiro. (My father-in-law is very cool.)
2. Vou almoçar a casa do meu sogro. (I am going to have lunch at my father-in-law's house.)
3. Eu dou-me muito bem com o meu sogro. (I get along very well with my father-in-law.)
4. O meu sogro deu-me um relógio de presente. (My father-in-law gave me a watch as a gift.)
5. Conheces o meu sogro? (Do you know my father-in-law?)