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

Yes

The spelling, meaning, and grammatical gender of "açúcar" are identical in both Brazilian and Continental Portuguese. The only differences lie in pronunciation:

• Vowels: Brazilian Portuguese tends to have more open and distinct vowel sounds. In Continental Portuguese, unstressed vowels are often more closed or even reduced to a neutral sound.
• The "r" sound: In many regions of Brazil, the final "r" is aspirated (sounding like an "h") or retroflex (the "caipira" R). In Portugal, the final "r" is typically a distinct alveolar tap or a more subtle, retracted sound.

Brazilian Portuguese Examples

1. Eu não tomo café com açúcar. (I don't drink coffee with sugar.)
2. Você pode me passar o açúcar, por favor? (Can you pass me the sugar, please?)
3. Esse bolo tem muito açúcar. (This cake has a lot of sugar.)
4. Eu estou tentando reduzir o açúcar na minha dieta. (I am trying to reduce sugar in my diet.)
5. O açúcar acabou, preciso comprar mais. (The sugar ran out, I need to buy more.)

Continental Portuguese Examples

1. Eu não tomo café com açúcar. (I don't drink coffee with sugar.)
2. Podes passar-me o açúcar, por favor? (Can you pass me the sugar, please?)
3. Este bolo tem muito açúcar. (This cake has a lot of sugar.)
4. Estou a tentar reduzir o açúcar na minha dieta. (I am trying to reduce sugar in my diet.)
5. O açúcar acabou, preciso de comprar mais. (The sugar ran out, I need to buy more.)