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

Yes

The meaning, grammar, and spelling of "espero" are identical in both Brazilian and Continental Portuguese. The only differences are in pronunciation. Brazilian Portuguese is generally more syllable-timed, meaning the vowels are more open and clearly articulated. In contrast, Continental Portuguese is more stress-timed, which leads to significant vowel reduction; an unstressed "e" or "o" in a word might be "swallowed" or pronounced much more subtly by a speaker from Portugal.

Brazilian Portuguese Examples

1. Espero que você chegue logo. (I hope you arrive soon.)
2. Espero o ônibus aqui todo dia. (I wait for the bus here every day.)
3. Espero que tudo dê certo com o seu trabalho. (I hope everything works out with your job.)
4. Não espero muito desse filme. (I don't expect much from this movie.)
5. Espero por você no shopping. (I am waiting for you at the mall.)

Continental Portuguese Examples

1. Espero que chegues brevemente. (I hope you arrive soon.)
2. Espero o autocarro aqui todos os dias. (I wait for the bus here every day.)
3. Espero que tudo corra bem no teu trabalho. (I hope everything goes well with your job.)
4. Não espero muito deste filme. (I don't expect much from this movie.)
5. Espero por ti no centro comercial. (I am waiting for you at the mall.)