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

No

While the dictionary definition of the word is the same, there are significant differences in usage and grammar.

In Brazilian Portuguese, "esquecer" is frequently used as a transitive direct verb, meaning you can say "esqueci a chave" (I forgot the key) without a reflexive pronoun. In Continental Portuguese, it is much more common and grammatally standard to use the pronominal/reflexive form "esquecer-se," which requires the preposition "de" (e.g., "esqueci-me da chave").

Additionally, there are pronunciation differences: Brazilian Portuguese is characterized by more open, musical vowels, whereas Continental Portuguese tends to have "closed" vowels and more frequent vowel reduction, making it sound more consonant-heavy.

Brazilian Portuguese Examples

1. Eu esqueci meu celular em casa. (I forgot my cell phone at home.)
2. Não esquece de me ligar amanhã. (Don't forget to call me tomorrow.)
3. Ele esqueceu o guarda-chuva no ônibus. (He forgot the umbrella on the bus.)
4. A gente esqueceu de trancar a porta. (We forgot to lock the door.)
5. Você esqueceu o que eu disse? (Did you forget what I said?)

Continental Portuguese Examples

1. Esqueci-me do telemóvel em casa. (I forgot my mobile phone at home.)
2. Não te esqueças de me ligar amanhã. (Don't forget to call me tomorrow.)
3. Ele esqueceu-se do guarda-chuva no autocarro. (He forgot the umbrella on the bus.)
4. Nós esquecemo-nos de trancar a porta. (We forgot to lock the door.)
5. Esqueceste-te do que eu disse? (Did you forget what I said?)