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

No

While the spelling and fundamental meaning of "onde" are identical in both dialects, there is a difference in typical usage. In Brazilian Portuguese, "onde" is frequently used colloquially in place of "aonde" when expressing movement (e.g., "Onde você vai?" instead of "Aonde você vai?"). In Continental Portuguese, the distinction between "onde" (for static locations) and "aonde" (for movement or direction) is much more strictly observed in standard speech. Additionally, Portuguese speakers often use the emphatic "é que" construction with "onde" in natural conversation.

Brazilian Portuguese

1. Onde você mora? (Where do you live?)
2. Onde fica o banheiro? (Where is the bathroom?)
3. Onde você vai agora? (Where are you going now?)
4. Não sei onde deixei minha chave. (I don't know where I left my key.)
5. Onde tem uma padaria por aqui? (Where is there a bakery around here?)

Continental Portuguese

1. Onde é que tu moras? (Where do you live?)
2. Onde fica a casa de banho? (Where is the bathroom?)
3. Aonde é que vais agora? (Where are you going now?)
4. Não sei onde deixei a minha chave. (I don't know where I left my key.)
5. Onde há uma pastelaria por aqui? (Where is there a bakery around here?)