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

No

While the word "quanto" maintains the same meaning, spelling, and grammatical category (interrogative pronoun/adjective or relative pronoun) in both dialects, there are differences in typical usage and pronunciation.

In Brazilian Portuguese, it is common to use the construction "quanto de" + [noun] (e.g., quanto de açúcar), whereas in Continental Portuguese, the preposition "de" is typically omitted (e.g., quanto açúcar). Additionally, the pronunciation differs significantly: Brazilian Portuguese features more open vowels and a more melodic, syllable-timed rhythm, while Continental Portuguese is much more "stress-timed," featuring heavy vowel reduction where unstressed vowels often become nearly silent.

Brazilian Portuguese Examples

1. Quanto custa isso? (How much does this cost?)
2. Quanto de açúcar você quer? (How much sugar do you want?)
3. Eu não sei quanto tempo vai demorar. (I don't know how much time it will take.)
4. Quanto de dinheiro você tem guardado? (How much money do you have saved?)
5. Você não imagina quanto eu gosto de você. (You can't imagine how much I like you.)

Continental Portuguese Examples

1. Quanto custa isto? (How much does this cost?)
2. Quanto açúcar queres? (How much sugar do you want?)
3. Não sei quanto tempo vai demorar. (I don't know how much time it will take.)
4. Quanto dinheiro tens guardado? (How much money do you have saved?)
5. Não imaginas quanto gosto de ti. (You can't imagine how much I like you.)