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

No

The difference lies in vocabulary and spelling. While the concept is identical, Brazilians use the word esporte, whereas the standard and most natural term in Portugal is desporto. Using "esporte" in Portugal is considered a Brazilianism and is much less common than the native European Portuguese term.

Brazilian Portuguese examples:

1. Eu amo praticar esporte todos os dias. (I love practicing sports every day.)
2. O futebol é o esporte mais popular do Brasil. (Soccer is the most popular sport in Brazil.)
3. Qual é o seu esporte favorito? (What is your favorite sport?)
4. Praticar esporte ajuda a reduzir o estresse. (Practicing sports helps to reduce stress.)
5. Ele começou a praticar um novo esporte este mês. (He started practicing a new sport this month.)

Continental Portuguese examples:

1. Eu adoro praticar desporto todos os dias. (I love practicing sports every day.)
2. O futebol é o desporto mais popular em Portugal. (Soccer is the most popular sport in Portugal.)
3. Qual é o teu desporto preferido? (What is your favorite sport?)
4. Praticar desporto ajuda a reduzir o stress. (Practising sports helps to reduce stress.)
5. Ele começou a praticar um novo desporto este mês. (He started practicing a new sport this month.)