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

No

The difference lies in typical usage and social connotation. In Brazil, "doutor" is used much more broadly as a social honorific. It is the standard way to address lawyers and physicians, but it is also frequently used as a polite or deferential way to address people of higher social status, or even as a respectful way to address strangers and elders in many regions. In Portugal, the use of "doutor" is much more restricted and formal, reserved almost exclusively for those who hold an academic doctorate (PhD) or a medical degree. When addressing a person of respect who does not hold these specific titles, a Portuguese person would typically use "senhor."

Brazilian Portuguese Usage

1. "O doutor já chegou ao consultório?" (Has the doctor arrived at the office?)
2. "Vou falar com o doutor sobre o processo judicial." (I am going to speak with the lawyer about the lawsuit.)
3. "Bom dia, doutor, pode me ajudar com esta sacola?" (Good morning, sir, can you help me with this bag?)
4. "Ele é doutor em biologia pela USP." (He has a PhD in biology from USP.)
5. "Não venha com essa história, doutor!" (Don't come at me with that story, "doctor"!)

Continental Portuguese Usage

1. "O médico já chegou ao consultório?" (Has the doctor arrived at the office?)
2. "Vou falar com o advogado sobre o processo judicial." (I am going to speak with the lawyer about the lawsuit.)
3. "Bom dia, senhor, pode ajudar-me com esta sacola?" (Good morning, sir, can you help me with this bag?)
4. "Ele é doutor em biologia pela USP." (He has a PhD in biology from USP.)
5. "Não me venhas com essa conversa!" (Don't come at me with that [argument]!)