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

No

While the core definition of "montar" (to assemble, to mount, or to set up) remains identical in both dialects, there are differences in typical usage. The word "montar" is used in different contexts because the vocabulary surrounding it changes. For example, the objects being "montados" often have different names (e.g., quebra-cabeça vs. puzzle). Furthermore, in Brazil, "montar" is a very common verb used to describe assembling an outfit or "look," whereas in Portugal, people would more naturally use verbs like compor (to compose) or escolher (to choose).

Brazilian Portuguese

1. Eu adoro montar quebra-cabeças de mil peças. (I love assembling thousand-piece puzzles.)
2. Ele aprendeu a montar na bicicleta muito cedo. (He learned to get on the bike very early.)
3. Vamos montar uma barraca para o acampamento. (We are going to set up a tent for the camping trip.)
4. Ela está montando um look para a festa. (She is putting together a look for the party.)
5. O marceneiro vai montar o guarda-roupa amanhã. (The carpenter is going to assemble the wardrobe tomorrow.)

Continental Portuguese

1. Eu adoro montar puzzles de mil peças. (I love assembling thousand-piece puzzles.)
2. Ele aprendeu a andar de bicicleta muito cedo. (He learned to ride a bike very early.)
3. Vamos montar uma tenda para o acampamento. (We are going to set up a tent for the camping trip.)
4. Ela vai compor um visual para a festa. (She is going to compose a look for the party.)
5. O carpinteiro vai montar o roupeiro amanhã. (The carpenter is going to assemble the wardrobe tomorrow.)