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

No.

While the fundamental meaning and spelling of "precisa" are identical in both dialects, the answer is No because there is a difference in typical usage (syntax). In Brazil, it is very common and natural to omit the preposition "de" when the verb precisar is followed by an infinitive (e.g., precisa fazer). In Portugal, the use of the preposition "de" is grammatically required and much more strictly observed (e.g., precisa de fazer). Additionally, there is a difference in pronunciation: Brazilian Portuguese features more open vowels, whereas Continental Portuguese features more closed, reduced vowels.

Brazilian Portuguese

1. Ela precisa comer agora. (She needs to eat now.)
2. Ele precisa de ajuda. (He needs help.)
3. Você precisa estudar para a prova. (You need to study for the test.)
4. O computador precisa arrumar. (The computer needs fixing.)
5. A resposta precisa ser exata. (The answer needs to be exact.)

Continental Portuguese

1. Ela precisa de comer agora. (She needs to eat now.)
2. Ele precisa de ajuda. (He needs help.)
3. Tu precisas de estudar para o exame. (You need to study for the exam.)
4. O computador precisa de ser arranjado. (The computer needs to be fixed.)
5. A resposta precisa de ser exata. (The answer needs to be exact.)