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

No

While the spelling and core meaning of "estudar" are identical in both varieties, there are significant differences in typical usage and pronksunciation:

• Usage (Continuous Aspect): In Brazil, the word is most commonly used in the gerund form (estudando) to express ongoing actions. In Portugal, the standard construction is the preposition a + the infinitive (a estudar).
• Usage (Syntax): In Brazil, verbs of necessity like precisar often omit the preposition (preciso estudar), whereas in Portugal, the preposition is typically retained (preciso de estudar).
• Pronunciation: In Continental Portuguese, the unstressed initial "e" often undergoes vowel reduction, sounding like a neutral, nearly silent schwa [ɨ], whereas in Brazilian Portuguese, it is more clearly pronounced as [e].

Brazilian Portuguese Examples:

1. Eu estou estudando muito para a prova. (I am studying a lot for the test.)
2. Você quer estudar comigo hoje à noite? (Do you want to study with me tonight?)
3. Nós precisamos estudar para o vestibular. (We need to study for the entrance exam.)
4. Eles estão estudando para o concurso. (They are studying for the civil service exam.)
5. Ela estuda na faculdade de direito. (She studies at the law school.)

Continental Portuguese Examples:

1. Eu estou a estudar muito para o exame. (I am studying a lot for the exam.)
2. Tu queres estudar comigo hoje à noite? (Do you want to study with me tonight?)
3. Nós precisamos de estudar para os exames. (We need to study for the exams.)
4. Eles estão a estudar para o concurso. (They are studying for the civil service exam.)
5. Ela estuda na faculdade de direito. (She studies at the law school.)