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

No

While the word "real" functions identically as an adjective (meaning "actual," "true," or "authentic") in both dialects, there is a difference in meaning when used as a noun. In Brazil, o real is the current official currency. In Portugal, o real refers to a historical currency that is no longer in circulation. Additionally, there is a difference in pronunciation: in Brazil, the "l" at the end of a syllable is typically vocalized as a "u" (sounding like re-au), whereas in Portugal, the "l" is a distinct alveolar sound.

Brazilian Portuguese Examples

1. Isso não é real, é um sonho. (This isn't real, it's a dream.)
2. Eu tenho cinquenta reais na carteira. (I have fifty reais in my wallet.)
3. Ele é um amigo real e leal. (He is a true and loyal friend.)
4. O valor real do imóvel é alto. (The real value of the property is high.)
5. Preciso de dez reais para o ônibus. (I need ten reais for the bus.)

Continental Portuguese Examples

1. Isto não é real, é um sonho. (This isn't real, it's a dream.)
2. Eu tenho cinquenta euros na carteira. (I have fifty euros in my wallet.)
3. Ele é um amigo de verdade e leal. (He is a true and loyal friend.)
4. O valor real do imóvel é elevado. (The real value of the property is high.)
5. Preciso de dez euros para o autocarro. (I need ten euros for the bus.)