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

No.

While the spelling and core meaning of "aprender" are identical in both dialects, there are differences in pronunciation and typical usage. In terms of pronunciation, Brazilian Portuguese tends to have more open, clear vowels, whereas Continental Portuguese features significant vowel reduction, where unstressed vowels are shortened or nearly silenced. Regarding usage, the primary difference lies in how the verb is used to express ongoing actions: Brazilians typically use the gerund (e.g., aprendendo), while Portuguese people use the construction "a" + infinitive (e.g., a aprender).

Brazilian Portuguese Examples

1. Eu estou aprendendo português. (I am learning Portuguese.)
2. Ele aprende muito rápido. (He learns very fast.)
3. Eu aprendi uma coisa nova hoje. (I learned something new today.)
4. Você quer aprender a tocar violão? (Do you want to learn to play the guitar?)
5. A gente está aprendendo as regras do jogo. (We are learning the rules of the game.)

Continental Portuguese Examples

1. Eu estou a aprender português. (I am learning Portuguese.)
2. Ele aprende muito rápido. (He learns very fast.)
3. Eu aprendi uma coisa nova hoje. (I learned something new today.)
4. Queres aprender a tocar guitarra? (Do you want to learn to play the guitar?)
5. Estamos a aprender as regras do jogo. (We are learning the rules of the game.)