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

Yes

The word "mostra" is semantically, grammatically, and orthographically identical in both Brazilian and Continental Portuguese. It functions as a noun (meaning an exhibition, fair, or display) and as the third-person singular present indicative of the verb mostrar (to show). The only significant difference is pronunciation. In Brazilian Portuguese, the vowels are generally more open and clearly articulated, whereas in Continental Portuguese, unstressed vowels tend to be reduced or "swallowed," making the word sound more clipped and the rhythm more stress-timed.

Brazilian Portuguese Examples

1. A mostra de cinema é muito famosa no Brasil. (The film festival is very famous in Brazil.)
2. Ele mostra o caminho para os turistas. (He shows the way to the tourists.)
3. A mostra de artesanato acontece todo ano. (The craft fair happens every year.)
4. Ela me mostrou o novo projeto. (She showed me the new project.)
5. A mostra de arte está aberta até as seis. (The art exhibition is open until six.)

Portuguese (Continental) Examples

1. A mostra de cinema é muito famosa em Portugal. (The film festival is very famous in Portugal.)
2. Ele mostra o caminho aos turistas. (He shows the way to the tourists.)
3. A feira de artesanato acontece todos os anos. (The craft fair happens every year.)
4. Ela mostrou-me o novo projeto. (She showed me the new project.)
5. A exposição de arte está aberta até às seis. (The art exhibition is open until six.)