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

Yes

The word "atravessar" is identical in spelling, meaning, and grammatical function in both Brazilian and European Portuguese. The only differences between the two dialects are phonetic. In Brazilian Portuguese, the pronunciation is generally more melodic and "syllable-timed," with more open vowels. In European Portuguese, the pronunciation is more "stress-timed" (or "clipped"), and unstressed vowels are often reduced, neutralized, or even swallowed almost entirely.

Brazilian Portuguese Examples

1. Eu vou atravessar a rua agora. (I am going to cross the street now.)
2. O motorista atravessou o sinal vermelho. (The driver ran a red light.)
3. Ela atravessou o oceano para nos visitar. (She crossed the ocean to visit us.)
4. Não deixe nada atravessar o seu caminho. (Don't let anything cross your path.)
5. Estamos atravessando uma fase difícil na empresa. (We are going through a difficult phase in the company.)

European Portuguese Examples

1. Vou atravessar a rua agora. (I am going to cross the street now.)
2. O condutor passou o sinal vermelho. (The driver ran the red light.)
3. Ela atravessou o oceano para nos visitar. (She crossed the ocean to visit us.)
4. Não deixes nada se atravessar no teu caminho. (Don't let anything cross your path.)
5. Estamos a atravessar uma fase difícil na empresa. (We are going through a difficult phase in the company.)