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

Yes.

The meaning, spelling, and grammatical function of "moderno" are identical in both Brazilian and European Portuguese. The only difference is pronunciation. Brazilian Portuguese features more open vowels and a more syllable-timed rhythm, whereas European Portuguese features much more closed, almost swallowed, unstressed vowels and a stress-timed rhythm (where unstressed syllables are significantly reduced).

Brazilian Portuguese Examples

1. Esse apartamento é muito moderno e bem iluminado. (This apartment is very modern and well-lit.)
2. A tecnologia moderna facilita muito a nossa vida. (Modern technology makes our lives much easier.)
3. Eu gosto de móveis com um design mais moderno. (I like furniture with a more modern design.)
4. O museu tem uma coleção de arte moderna incrível. (The museum has an incredible modern art collection.)
5. Ele sempre tenta usar as roupas mais modernas que encontra. (He always tries to use the most modern clothes he finds.)

European Portuguese Examples

1. Este apartamento é muito moderno e bem iluminado. (This apartment is very modern and well-lit.)
2. A tecnologia moderna facilita muito a nossa vida. (Modern technology makes our lives much easier.)
3. Gosto de móveis com um design mais moderno. (I like furniture with a more modern design.)
4. O museu tem uma coleção de arte moderna incrível. (The museum has an incredible modern art collection.)
5. Ele tenta sempre usar as roupas mais modernas que encontra. (He always tries to use the most modern clothes he finds.)