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

Yes

The word "pássaro" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference is pronunciation. Brazilian Portuguese generally features more open, clearly articulated vowels. In Continental Portuguese, unstressed vowels tend to be "closed" or reduced, which gives the word a more clipped and staccato sound.

Brazilian Portuguese

1. Eu vi um pássaro azul no jardim. (I saw a blue bird in the garden.)
2. O pássaro está cantando muito cedo. (The bird is singing very early.)
3. Aquele pássaro é muito bonito. (That bird is very beautiful.)
4. Eu gosto de observar pássaros na floresta. (I like to observe birds in the forest.)
5. O pássaro voou para longe. (The bird flew far away.)

Continental Portuguese

1. Eu vi um pássaro azul no jardim. (I saw a blue bird in the garden.)
2. O pássaro está a cantar muito cedo. (The bird is singing very early.)
3. Aquele pássaro é muito bonito. (That bird is very beautiful.)
4. Gosto de observar pássaros na floresta. (I like to observe birds in the forest.)
5. O pássaro voou para longe. (The bird flew far away.)