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

Yes

The word "médico" is identical in Brazilian and Continental Portuguese regarding its meaning, spelling, and grammatical function. The primary difference is pronunciation. In Brazilian Portuguese, the vowels are typically more open and the rhythm is syllable-timed, making the "e" and "i" sounds more distinct. In Continental Portuguese, unstressed vowels are often reduced or "closed"; for example, the "i" in "médico" is often much shorter or almost swallowed, and the "e" can sound slightly different depending on the regional accent.

Brazilian Portuguese Examples

1. O médico me atendeu muito bem. (The doctor treated me very well.)
2. Eu preciso marcar uma consulta com o médico. (I need to schedule an appointment with the doctor.)
3. O médico deu a receita para o meu remédio. (The doctor gave the prescription for my medicine.)
4. Ela é uma médica muito dedicada. (She is a very dedicated doctor.)
5. O médico está no hospital agora. (The doctor is at the hospital now.)

Continental Portuguese Examples

1. O médico atendeu-me muito bem. (The doctor treated me very well.)
2. Eu preciso de marcar uma consulta com o médico. (I need to schedule an appointment with the doctor.)
3. O médico deu a receita para o meu medicamento. (The doctor gave the prescription for my medication.)
4. Ela é uma médica muito dedicada. (She is a very dedicated doctor.)
5. O médico está no hospital agora. (The doctor is at the hospital now.)