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

Yes

The word "outro" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels tend to be more open and the rhythm is more "syllable-timed," making the diphthong "ou" sound clear and melodic. In Continental Portuguese, vowels are often reduced or "swallowed" (unaccented vowels become much shorter and more closed), which can make the word sound much more clipped and shorter to a non-native ear.

Brazilian Portuguese Examples

1. Eu quero outro café, por favor. (I want another coffee, please.)
2. Tem outro jeito de fazer isso? (Is there another way to do this?)
3. Eu não achei outro livro na estante. (I didn't find another book on the shelf.)
4. Outra vez você esqueceu a chave! (Another time [Once again] you forgot the key!)
5. Vamos para outro lugar hoje. (Let's go to another place today.)

Continental Portuguese Examples

1. Eu queria outro café, por favor. (I would like another coffee, please.)
2. Há outro modo de fazer isto? (Is there another way to do this?)
3. Não encontrei outro livro na estante. (I didn't find another book on the shelf.)
4. Outra vez esqueceste-te da chave! (Another time you forgot the key!)
5. Vamos para outro sítio hoje. (Let's go to another place today.)