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

No

While the spelling, grammar, and core meaning of "remédio" are identical in both dialects, there is a difference in typical usage and connotation. In Brazil, "remédio" is the universal, everyday term used for all types of medication, whether they are pharmaceutical drugs or natural treatments. In Portugal, while "remédio" is used, it often carries a connotation of "home remedies" (remédios caseiros) or natural/folk medicine. For pharmaceutical products purchased at a pharmacy, a Portuguese person is much more likely to use the word medicamento. Additionally, the pronunciation differs: Brazilians typically use a palatalized "d" (sounding like "j"), whereas the Portuguese use a dental "d."

Brazilian Portuguese Usage

1. "Eu esqueci de tomar o meu remédio." (I forgot to take my medicine.)
2. "Esse remédio é muito forte para uma criança." (This medicine is very strong for a child.)
3. "Você precisa comprar esse remédio na farmácia?" (Do you need to buy this medicine at the pharmacy?)
4. "Não tem remédio para essa dor." (There is no medicine for this pain.)
5. "O médico passou um remédio novo para mim." (The doctor prescribed a new medicine for me.)

Continental Portuguese Usage

1. "Esqueci-me de tomar o meu medicamento." (I forgot to take my medicine.)
2. "Este medicamento é muito forte para uma criança." (This medicine is very strong for a child.)
3. "Precisas de comprar este medicamento na farmácia?" (Do you need to buy this medicine at the pharmacy?)
4. "Não há remédio para essa dor." (There is no remedy for this pain.)
5. "O médico receitou-me um medicamento novo." (The doctor prescribed a new medicine for me.)