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

Yes

The word "claro" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the final "o" is heavily reduced, sounding almost like a "u" ([ˈkla.ɾu]). In Continental Portuguese, while the final "o" is also reduced, the phonetic realization and the rhythmic stress of the word differ due to the more closed vowel sounds and the distinct prosody (cadence) characteristic of the European dialect.

Brazilian Portuguese Examples

1. Claro que eu vou à festa! (Of course I'm going to the party!)
2. Eu prefiro azul claro. (I prefer light blue.)
3. Isso está muito claro para você. (That is very clear to you.)
4. Claro, eu te ajudo com isso. (Sure, I'll help you with that.)
5. O dia está bem claro hoje. (The day is very bright/clear today.)

Continental Portuguese Examples

1. Claro que eu vou à festa! (Of course I'm going to the party!)
2. Prefiro azul claro. (I prefer light blue.)
3. Isto está muito claro para ti. (This is very clear to you.)
4. Claro, eu ajudo-te com isso. (Sure, I'll help you with that.)
5. O dia está muito claro hoje. (The day is very bright/clear today.)