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

Yes

The word "curiosidade" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only differences lie in pronunciation:

• Vowel Quality: In Brazilian Portuguese, the vowels are generally more "open" and clearly articulated. In Continental Portuguese, unstressed vowels are much more "closed" or reduced, often sounding almost swallowed.
• The "d" sound: In most regions of Brazil, the "d" in "curiosidade" is pronounced as an affricate [dʒ], sounding like the "j" in the English word "jeep" (curio-si-da-de $\rightarrow$ curio-si-ja-de). In Portugal, the "d" remains a plain dental [d], as in the English word "dog."

Brazilian Portuguese Examples

1. "Sinto muita curiosidade sobre esse assunto." (I feel much curiosity about that subject.) **2. "Aquele museu tem muitas curiosidades para mostrar." (That museum has many curiosities to show.)
2. "Ele é um garoto muito curioso." (He is a very curious boy.)
3. "Cuidado, a curiosidade matou o gato!" (Careful, curiosity killed the cat!)
4. "Por pura curiosidade, você sabe onde fica a padaria?" (Out of pure curiosity, do you know where the bakery is?)

Continental Portuguese Examples

1. "Tenho uma curiosidade enorme sobre este tema." (I have an enormous curiosity about this theme.)
2. "Este casarão é uma curiosidade da nossa região." (This mansion is a curiosity of our region.)
3. "Ele é um rapaz muito curioso." (He is a very curious young man.)
4. "Não te metas em tudo, a curiosidade mata o gato!" (Don't meddle in everything, curiosity kills the cat!)
5. "Só por curiosidade, sabes se o autocarro já passou?" (Just out of curiosity, do you know if the bus has already passed?)