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

Yes

The spelling, meaning, and grammatical function of "coelho" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. In Brazilian Portuguese, the unstressed "e" is typically pronounced more openly and clearly. In Continental Portuguese, unstressed vowels are heavily reduced, meaning the "e" is often swallowed or pronounced with a very closed, muffled sound.

Brazilian Portuguese Examples

1. O coelho é muito fofinho. (The rabbit is very cute.)
2. Eu comprei um coelho de estimação. (I bought a pet rabbit.)
3. O coelho está comendo uma cenoura. (The rabbit is eating a carrot.)
4. Meu cachorro viu o coelho no jardim. (My dog saw the rabbit in the garden.)
5. O coelho da Páscoa é muito legal. (The Easter Bunny is very cool.)

Continental Portuguese Examples

1. O coelho é muito fofo. (The rabbit is very cute.)
2. Comprei um coelho de estimação. (I bought a pet rabbit.)
3. O coelho está a comer uma cenoura. (The rabbit is eating a carrot.)
4. O meu cão viu o coelho no jardim. (My dog saw the rabbit in the garden.)
5. O coelho da Páscoa é muito fixe. (The Easter Bunny is very cool.)