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

No

While the core meaning of "passeio" (an outing, a stroll, or an excursion) is identical in both dialects, there is a difference in typical usage and meaning regarding urban infrastructure. In Portugal, "passeio" is the standard term used to refer to the sidewalk or pavement. In Brazil, that specific meaning is virtually non-existent in daily conversation; instead, Brazilians use the word "calçada." Therefore, the word carries an extra semantic layer in Continental Portuguese that it does not have in Brazilian Portuguese.

Brazilian Portuguese Examples

1. Vou dar um passeio no parque amanhã. (I am going to take a walk in the park tomorrow.)
2. O cachorro adorou o passeio. (The dog loved the walk.)
3. A prova de matemática foi um passeio. (The math exam was a breeze.)
4. Eu tropecei na calçada enquanto corria. (I tripped on the sidewalk while running.)
5. O passeio de barco foi muito legal. (The boat trip was very cool.)

Continental Portuguese Examples

1. Vou dar um passeio no parque amanhã. (I am going to take a walk in the park tomorrow.)
2. O cão adorou o passeio. (The dog loved the walk.)
3. O exame de matemática foi um passeio. (The math exam was a breeze.)
4. Eu tropecei no passeio enquanto corria. (I tripped on the sidewalk while running.)
5. O passeio de barco foi muito agradável. (The boat trip was very pleasant.)