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

No

While the fundamental meaning and spelling of "esperar" are identical in both dialects, there are significant differences in pronunciation and typical usage/syntax. In Brazilian Portuguese, vowels are generally more open and pronounced, whereas in Continental Portuguese, vowels are often reduced or nearly silent, creating a more "consonantal" sound. Additionally, while the verb "esperar" can be used in both, a Portuguese person frequently uses the locution "estar à espera de" to express the act of waiting, and the grammatical placement of pronouns (clitics) differs significantly.

Brazilian Portuguese Usage

1. Estou esperando o ônibus. (I am waiting for the bus.)
2. Espero que você consiga. (I hope you can do it.)
3. Não me espera, eu vou demorar. (Don't wait for me, I'll be late.)
4. A gente está esperando o resultado. (We are waiting for the result.)
5. Espero que tudo dê certo. (I hope everything works out.)

Continental Portuguese Usage

1. Estou à espera do autocarro. (I am waiting for the bus.)
2. Espero que consigas. (I hope you can do it.)
3. Não me esperes, eu vou demorar. (Don't wait for me, I'll be late.)
4. Estamos à espera do resultado. (We are waiting for the result.)
5. Espero que corra tudo bem. (I hope everything goes well.)