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

Yes

The word "nadar" is identical in spelling, meaning, and grammatical classification in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are generally more open and the rhythm is more melodic (syllable-timed). In Continental Portuguese, the pronunciation is more "stress-timed," meaning unstressed vowels are often reduced, shortened, or almost silent, creating a more "clipped" sound.

Brazilian Portuguese

1. Eu gosto de nadar no mar. (I like to swim in the sea.)
2. Ele está nadando na piscina agora. (He is swimming in the pool right now.)
3. Você sabe nadar bem? (Do you know how to swim well?)
4. As crianças estão aprendendo a nadar. (The children are learning to swim.)
5. Eu vou nadar um pouco para relaxar. (I am going to swim a little to relax.)

Continental Portuguese

1. Eu gosto de nadar no mar. (I like to swim in the sea.)
2. Ele está a nadar na piscina agora. (He is swimming in the pool right now.)
3. Tu sabes nadar bem? (Do you know how to swim well?)
4. As crianças estão a aprender a nadar. (The children are learning to swim.)
5. Vou nadar um pouco para relaxar. (I am going to swim a little to relax.)