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

No

While the fundamental meaning of the animal is the same, there is a significant difference in typical usage. In Brazilian Portuguese, cachorro is the standard, universal word used for any dog, regardless of its age or size. In Continental Portuguese, the standard word for a dog is cão. In Portugal, the word cachorro is much more commonly used to refer specifically to a puppy (filhote) or a very small breed. Using "cachorro" to refer to a large, adult dog in Portugal may sound slightly less natural than using "cão."

Brazilian Portuguese Examples

1. O meu cachorro é muito brincalhão. (My dog is very playful.)
2. Você viu aquele cachorro na rua? (Did you see that dog in the street?)
3. Eu adoro passear com o meu cachorro. (I love walking with my dog.)
4. O cachorro do vizinho late muito à noite. (The neighbor's dog barks a lot at night.)
5. Ela comprou comida para o cachorro. (She bought food for the dog.)

Continental Portuguese Examples

1. O meu cão é muito brincalhão. (My dog is very playful.)
2. Viste aquele cão na rua? (Did you see that dog in the street?)
3. Adoro passear com o meu cão. (I love walking with my dog.)
4. O cão do vizinho ladra muito à noite. (The neighbor's dog barks a lot at night.)
5. Ela comprou comida para o cão. (She bought food for the dog.)