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

Yes

The word "natação" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. The only difference is in pronunciation. In Brazil, the vowels tend to be more open, and the nasal diphthong "ão" is pronounced with a more resonant, open nasal quality. In Portugal, the pronunciation is more "closed," and speakers often exhibit more significant vowel reduction in unstressed syllables, making the word sound more "staccato" or clipped to a Brazilian ear.

Brazilian Portuguese Examples:

1. Eu faço natação todas as manhãs. (I do swimming every morning.)
2. A aula de natação é muito legal. (The swimming class is very cool.)
3. A natação é o meu esporte favorito. (Swimming is my favorite sport.)
4. Eu comprei um traje de natação novo. (I bought a new swimming suit.)
5. Ele treina natação para as competições. (He trains swimming for competitions.)

Portuguese (Portugal) Examples:

1. Eu faço natação todas as manhãs. (I do swimming every morning.)
2. A aula de natação é muito fixe. (The swimming class is very cool.)
3. A natação é o meu desporto favorito. (Swimming is my favorite sport.)
4. Eu comprei um fato de banho novo para a natação. (I bought a new swimsuit for swimming.)
5. Ele treina natação para os campeonatos. (He trains swimming for championships.)