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

Yes

The word "carne" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only differences are phonetic. In Brazilian Portuguese, the final "e" is typically pronounced as a closed "i" (sounding like car-ni). In Continental Portuguese, the final "e" is a near-silent, unrounded, or very short vowel (/ɨ/), making the word sound more clipped. Additionally, the "r" in Brazil is often pronounced with a more retroflex or alveolar sound (depending on the region), whereas in Portugal, it is more likely to be pronounced as a uvular or guttural sound.

Brazilian Portuguese

1. Eu gosto de comer carne assada no domingo. (I like to eat roasted meat on Sundays.)
2. A carne está muito dura hoje. (The meat is very tough today.)
3. Vou comprar carne no açougue daqui a pouco. (I am going to buy meat at the butcher shop in a little while.)
4. Eu prefiro comer carne moída com batatas. (I prefer to eat ground meat with potatoes.)
5. Não como carne vermelha porque sou vegetariano. (I don't eat red meat because I am vegetarian.)

Portuguese (Continental)

1. Eu gosto de comer carne assada ao domingo. (I like to eat roasted meat on Sundays.)
2. A carne está demasiado salgada. (The meat is too salty.)
3. Vou comprar carne ao talho logo à tarde. (I am going to buy meat at the butcher shop later this afternoon.)
4. Eu prefiro comer carne picada com batatas. (I prefer to eat minced meat with potatoes.)
5. Não costumo comer carne de porco. (I don't usually eat pork.)