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

Yes.

The spelling, meaning, and grammatical function of "instrumento" are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, vowels are typically more open and syllables are pronounced with more clarity and distinctness. In Continental Portuguese, unstressed vowels undergo significant reduction; the final "o" is pronounced as a closed "u" [u], and unstressed vowels within the word are often shortened or nearly swallowed, creating a more "compressed" or "clipped" sound.

Brazilian Portuguese Examples

1. Eu estou aprendendo a tocar um novo instrumento. (I am learning to play a new instrument.)
2. A música é um instrumento que ajuda a relaxar. (Music is an instrument that helps to relax.)
3. Esse instrumento médico é muito caro. (This medical instrument is very expensive.)
4. A educação é o melhor instrumento de mudança. (Education is the best instrument of change.)
5. Ele usa a caneta como um instrumento de escrita. (He uses the pen as a writing instrument.)

Continental Portuguese Examples

1. Estou a aprender a tocar um novo instrumento. (I am learning to play a new instrument.)
2. A música é um instrumento que ajuda no relaxamento. (Music is an instrument that helps with relaxation.)
3. Este instrumento médico é muito dispendioso. (This medical instrument is very expensive.)
4. A educação é o principal instrumento de mudança. (Education is the main instrument of change.)
5. Ele utiliza a caneta como instrumento de escrita. (He uses the pen as a writing instrument.)