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

Yes

The meaning, spelling, and grammatical function of "caro" are identical in both Brazilian and Continental Portuguese. The only difference is pronunciation. Brazilian Portuguese tends to pronounce vowels more openly and clearly, whereas Continental Portuguese often reduces unstressed vowels, making the word sound more closed or clipped.

Brazilian Portuguese Examples

1. Esse celular é muito caro. (This cell phone is very expensive.)
2. Meu caro amigo, como você está? (My dear friend, how are you?)
3. O preço da carne está caro demais. (The price of meat is too expensive.)
4. Não achei esse presente muito caro. (I didn't find this gift very expensive.)
5. Caro cliente, sua fatura está disponível. (Dear customer, your invoice is available.)

Continental Portuguese Examples

1. Este telemóvel é muito caro. (This mobile phone is very expensive.)
2. Meu caro amigo, como estás? (My dear friend, how are you?)
3. O preço da carne está demasiado caro. (The price of meat is too expensive.)
4. Não achei este presente muito caro. (I didn't find this gift very expensive.)
5. Caro cliente, a sua fatura está disponível. (Dear customer, your invoice is available.)