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

No.

While the word is spelled the same and shares the same core meaning, there are differences in typical usage and grammar. In Brazil, "melhora" is commonly used as a singular noun to describe any type of improvement (e.g., "a melhora do tempo" — the improvement of the weather). In Portugal, "melhoria" is the much more frequent noun for general improvements, while "melhora" is specifically associated with health recovery. Furthermore, the idiomatic expression for "Get well" differs: Brazilians typically use the singular/interjection "Melhoras!", whereas the Portuguese use the definite article and the plural: "As melhoras!".

Brazilian Portuguese Usage

1. "Desejo uma rápida melhora para você." (I wish you a speedy recovery.)
2. "Notei uma melhora no seu desempenho." (I noticed an improvement in your performance.)
3. "Melhoras! Espero que você fique bem." (Get well! I hope you feel better.)
4. "A melhora do paciente surpreendeu os médicos." (The patient's improvement surprised the doctors.)
5. "Não vi melhora no quadro clínico." (I didn't see an improvement in the clinical condition.)

Continental Portuguese Usage

1. "Desejo-te as rápidas melhoras." (I wish you a speedy recovery.)
2. "Houve uma grande melhoria no serviço." (There was a great improvement in the service.)
3. "As melhoras! Espero que fiques bem." (Get well! I hope you feel better.)
4. "A melhoria do tempo foi muito positiva." (The improvement in the weather was very positive.)
5. "Desejo-te as melhoras após a cirurgia." (I wish you a recovery after the surgery.)