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

Yes

The word "sogra" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only differences are phonetic. In Brazilian Portuguese, the vowels tend to be more open and the pronunciation is more melodic/syllable-timed. In Continental Portuguese, unstressed vowels are often more closed or even elided (reduced), and the rhythm of the word is more stress-timed.

Brazilian Portuguese Examples

1. Minha sogra é muito legal. (My mother-in-law is very nice.)
2. Eu estou indo visitar minha sogra. (I am going to visit my mother-in-law.)
3. Minha sogra me ajudou ontem. (My mother-in-law helped me yesterday.)
4. Eu adoro a comida da minha sogra. (I love my mother-in-law's food.)
5. Minha sogra mora aqui perto. (My mother-in-law lives near here.)

Continental Portuguese Examples

1. A minha sogra é muito simpática. (My mother-in-law is very nice.)
2. Estou a ir visitar a minha sogra. (I am going to visit my mother-in-law.)
3. A minha sogra ajudou-me ontem. (My mother-in-law helped me yesterday.)
4. Adoro a comida da minha sogra. (I love my mother-in-law's food.)
5. A minha sogra mora aqui perto. (My mother-in-law lives near here.)