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

Yes

The word "avô" is identical in meaning, spelling, and grammar in both Brazilian and European Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are generally more open and clearly articulated. In Continental (European) Portuguese, the pronunciation tends to be more "closed" due to the reduction of unstressed vowels in speech, which can make the rhythm of the word sound different when embedded in a sentence.

Brazilian Portuguese

1. Meu avô é muito legal. (My grandfather is very cool.)
2. Eu vou visitar meu avô amanhã. (I am going to visit my grandfather tomorrow.)
3. O meu avô me contou uma história. (My grandfather told me a story.)
4. Meu avô me deu um presente. (My grandfather gave me a gift.)
5. A comida do meu avô é a melhor de todas. (My grandfather's food is the best of all.)

European Portuguese

1. O meu avô é muito porreiro. (My grandfather is very cool.)
2. Vou visitar o meu avô amanhã. (I am going to visit my grandfather tomorrow.)
3. O meu avô contou-me uma história. (My grandfather told me a story.)
4. O meu avô deu-me um presente. (My grandfather gave me a gift.)
5. A comida do meu avô é a melhor de todas. (My grandfather's food is the best of all.)