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

No.

While the word "presente" maintains the same fundamental meanings, grammar, and spelling in both Brazilian and Continental Portuguese, there are differences in typical usage and pronunciation:

• Usage: In Brazil, "presente" is the standard and most common word for a gift. In Portugal, while "presente" is perfectly correct, speakers often prefer the word "oferta" when referring to something given as a gesture or something received for free, and "lembrança" when referring to a small souvenir or a token of affection.
• Pronunciation: There is a significant phonetic difference in the unstressed final "e." In Brazilian Portuguese, this is typically pronounced as a clear /i/ (like the "ee" in "see"). In Continental Portuguese, the final "e" is a "reduced" vowel, sounding much more closed, like a muffled /ɨ/ (similar to the "u" in "but," but even more swallowed).

Brazilian Portuguese Examples

1. Eu comprei um presente para você. (I bought a gift for you.)
2. O presente de aniversário dele foi um livro. (His birthday gift was a book.)
3. Ele não estava presente na reunião de ontem. (He was not present at yesterday's meeting.)
4. O presente momento exige muita paciência. (The present moment requires a lot of patience.)
5. Ela é uma mãe muito presente na vida dos filhos. (She is a very present/involved mother in her children's lives.)

Continental Portuguese Examples

1. Eu recebi uma oferta de aniversário. (I received a birthday gift.)
2. Comprei uma lembrança para ti. (I bought a souvenir/small gift for you.)
3. Ele não estava presente na reunião de ontem. (He was not present at yesterday's meeting.)
4. O presente momento exige muita paciência. (The present moment requires a lot of patience.)
5. Ela é uma mãe muito presente na vida dos filhos. (She is a very present/involved mother in her children's lives.)