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

Yes

The word "boxe" refers to the sport of boxing in both Brazilian and Continental Portuguese. There are no differences in meaning, spelling, or grammatical category (it is a masculine noun). The only difference lies in the pronunciation, as the phonetic realization of the vowels and the rhythm of the word will vary according to the specific regional accent of the speaker.

Brazilian Portuguese Examples

1. Eu adoro assistir a lutas de boxe. (I love watching boxing matches.)
2. Ele começou a treinar boxe ontem. (He started training boxing yesterday.)
3. O boxe é um esporte muito intenso. (Boxing is a very intense sport.)
4. Minha academia tem aulas de boxe de manhã. (My gym has boxing classes in the morning.)
5. Eu comprei um par de luvas de boxe novas. (I bought a pair of new boxing gloves.)

Continental Portuguese Examples

1. Eu adoro ver lutas de boxe. (I love watching boxing matches.)
2. Ele começou a praticar boxe ontem. (He started practicing boxing yesterday.)
3. O boxe é um desporto muito intenso. (Boxing is a very intense sport.)
4. O meu ginásio tem aulas de boxe de manhã. (My gym has boxing classes in the morning.)
5. Eu comprei um par de luvas de boxe novas. (I bought a pair of new boxing gloves.)