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

No

While the spelling and core meaning of "avó" are identical in both varieties, there are significant differences in grammar and typical usage. Grammatically, Brazilians typically omit the definite article before possessive pronouns (e.g., "Minha avó"), whereas in Portugal, the article is almost always required (e.g., "A minha avó"). Regarding typical usage, Brazilians frequently use the diminutive "vovó" as the standard affectionate term, whereas in Portugal, "avó" remains the standard, with "avozinha" used for endearment.

Brazilian Portuguese

1. Minha avó é muito legal. (My grandmother is very cool.)
2. Eu adoro minha avó. (I adore my grandmother.)
3. Minha avó faz um bolo ótimo. (My grandmother makes a great cake.)
4. Vou visitar minha avó hoje. (I am going to visit my grandmother today.)
5. Minha avó mora longe. (My grandmother lives far away.)

Continental Portuguese

1. A minha avó é muito simpática. (My grandmother is very nice.)
2. Eu adoro a minha avó. (I adore my grandmother.)
3. A minha avó faz um bolo ótimo. (My grandmother makes a great cake.)
4. Vou visitar a minha avó hoje. (I am going to visit my grandmother today.)
5. A minha avó mora longe. (My grandmother lives far away.)