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

Yes.

The word "marcar" is identical in meaning, spelling, and grammar in both Brazilian and Continental Portuguese. It retains all its core functions: to schedule/book, to mark/label, to score (a goal), to dial (a number), and to tag (on social media). The differences observed between the two dialects occur in the contextual vocabulary (the nouns and objects the verb interacts with) and the syntax (how pronouns are placed in the sentence).

Brazilian Portuguese

1. Eu vou marcar um médico para amanhã. (I'm going to schedule a doctor for tomorrow.)
2. Ele marcou um gol incrível! (He scored an incredible goal!)
3. Você pode me marcar na foto? (Can you tag me in the photo?)
4. Não esqueça de marcar esse compromisso na agenda. (Don't forget to mark/note this appointment in the planner.)
5. Eu preciso marcar o número no celular. (I need to dial the number on the cell phone.)

Continental Portuguese

1. Eu vou marcar uma consulta para amanhã. (I'm going to schedule an appointment for tomorrow.)
2. Ele marcou um golo incrível! (He scored an incredible goal!)
3. Podes marcar-me na fotografia? (Can you tag me in the photograph?)
4. Não te esqueças de marcar esse compromisso na agenda. (Don't forget to mark/note this appointment in the planner.)
5. Eu preciso de marcar o número no telemóvel. (I need to dial the number on the mobile phone.)