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

No.

While the meaning and spelling of "mostrar" are identical in both dialects, there are significant differences in grammar and typical usage regarding how the word interacts with other parts of a sentence:

• Pronoun Placement (Syntax): In Brazilian Portuguese (BP), it is extremely common and natural to place object pronouns before the verb (e.g., "me mostrar"). In Continental Portuguese (CP), this is considered grammatically incorrect in standard usage; the pronoun is typically placed after the verb (e.g., "mostrar-me").
• The Continuous Aspect: To express an action currently in progress, Brazilians use the gerund ("mostrando"). Portuguese people use the preposition a + the infinitive ("a mostrar").
• Pronunciation: In Brazil, vowels tend to be more open and clearly articulated. In Portugal, unstressed vowels are often "reduced" or almost swallowed, and the "r" sound is typically more alveolar or tapped.

Brazilian Portuguese Examples

1. Você pode me mostrar o caminho? (Can you show me the way?)
2. Ele está mostrando o carro novo para os amigos. (He is showing the new car to his friends.)
3. Eu vou te mostrar o que eu comprei. (I am going to show you what I bought.)
4. Ela me mostrou a foto da viagem. (She showed me the photo from the trip.)
5. Me mostra como usar esse aplicativo? (Show me how to use this app?)

Continental Portuguese Examples

1. Podes mostrar-me o caminho? (Can you show me the way?)
2. Ele está a mostrar o automóvel novo aos amigos. (He is showing the new car to his friends.)
3. Eu vou mostrar-te o que comprei. (I am going to show you what I bought.)
4. Ela mostrou-me a fotografia da viagem. (She showed me the photograph from the trip.)
5. Mostra-me como se faz isto. (Show me how this is done.)