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

Yes

The differences in the word "natureza" are strictly phonetic. In Brazilian Portuguese, vowels are generally more open and clearly articulated, which gives the word a more resonant, "musical" sound. In Continental Portuguese, unstressed vowels tend to be more closed or even reduced (almost swallowed), which results in a more "staccato" or consonant-heavy pronunciation. There are no differences in spelling, grammar, or fundamental meaning.

Brazilian Portuguese Examples

1. Eu amo passar o tempo em contato com a natureza. (I love spending time in contact with nature.)
2. A natureza desse problema é muito complexa. (The nature of this problem is very complex.)
3. Ele tem uma natureza muito calma. (He has a very calm nature.)
4. A gente precisa preservar a natureza. (We need to preserve nature.)
5. A natureza humana é cheia de surpresas. (Human nature is full of surprises.)

Continental Portuguese Examples

1. Eu adoro passar o tempo em contacto com a natureza. (I love spending time in contact with nature.)
2. A natureza deste problema é muito complexa. (The nature of this problem is very complex.)
3. Ele tem uma natureza muito calma. (He has a very calm nature.)
4. Nós precisamos de preservar a natureza. (We need to preserve nature.)
5. A natureza humana é cheia de surpresas. (Human nature is full of surprises.)