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

Yes

The only difference in the use of the word "bigode" between Brazilian and Continental Portuguese is pronunciation. In Brazil, the pronunciation tends to be more open and the vowels are more clearly articulated (syllable-timed rhythm). In Portugal, the unstressed vowels are more closed or even reduced, and the rhythm is more stress-timed, often making the word sound shorter or more clipped to a non-native ear. The meaning, spelling, and grammatical function remain identical.

Brazilian Portuguese Examples

1. Ele resolveu deixar o bigode crescer. (He decided to let his mustache grow.)
2. Eu não gosto muito desse tipo de bigode. (I don't like this type of mustache very much.)
3. O bigode dele está precisando de uma aparada. (His mustache needs a trim.)
4. Ele usa um bigode bem grande e estiloso. (He wears a very big and stylish mustache.)
5. Aquele rapaz ficou muito diferente com esse bigode. (That guy looked very different with that mustache.)

Continental Portuguese Examples

1. Ele decidiu deixar o bigode crescer. (He decided to let his mustache grow.)
2. Eu não gosto muito deste tipo de bigode. (I don't like this type of mustache very much.)
3. O bigode dele precisa de uma poda. (His mustache needs a trim.)
4. Ele usa um bigode muito grande e elegante. (He wears a very big and elegant mustache.)
5. Esse bigode não te fica nada bem. (That mustache does not suit you at all.)