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

No

While the spelling, meaning, and grammatical identity of "vou" are identical in both dialects, the answer is No because there are significant differences in typical usage and pronunciation. In terms of usage, the prepositions that follow "vou" differ: Brazilians typically use the preposition em (e.g., vou no/na), whereas Portuguese people use the preposition a (e.g., vou ao/à). Additionally, the placement of pronouns following the verb and the construction of the continuous tense (using the gerund in Brazil vs. the infinitive in Portugal) differ. Pronunciation also varies, as the vowel sound is generally more closed in Continental Portuguese.

Brazilian Portuguese Usage

1. Eu vou no shopping. (I am going to the mall.)
2. Eu vou na sua casa amanhã. (I am going to your house tomorrow.)
3. Eu vou te ajudar com isso. (I am going to help you with this.)
4. Eu vou estar fazendo o jantar. (I will be making dinner.)
5. Eu vou no médico agora. (I am going to the doctor now.)

Continental Portuguese Usage

1. Eu vou ao shopping. (I am going to the mall.)
2. Eu vou à tua casa amanhã. (I am going to your house tomorrow.)
3. Eu vou ajudar-te com isso. (I am going to help you with this.)
4. Eu vou estar a fazer o jantar. (I will be making dinner.)
5. Eu vou ao médico agora. (I am going to the doctor now.)