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

No.

While the spelling, grammar, and core meaning of "ver" (to perceive with the eyes) are identical in both dialects, there is a difference in typical usage. In Brazil, "ver" is the standard and most common verb used to mean "to watch" (e.g., for movies, television, or videos). In Portugal, while "ver" is used for physical sight, the verb "assistir" is much more naturally preferred when referring to the consumption of media.

Brazilian Portuguese Usage

1. Eu vi um filme ontem. (I watched a movie yesterday.)
2. Você viu o jogo de ontem? (Did you see yesterday's game?)
3. Eu quero ver esse vídeo no YouTube. (I want to watch this video on YouTube.)
4. Vamos ver se o ônibus já passou. (Let's see if the bus has already passed.)
5. Eu vi você no shopping. (I saw you at the mall.)

Continental Portuguese Usage

1. Eu assisti a um filme ontem. (I watched a movie yesterday.)
2. Viste o jogo de ontem? (Did you see yesterday's game?)
3. Quero ver este vídeo no YouTube. (I want to watch this video on YouTube.)
4. Vamos ver se o autocarro já passou. (Let's see if the bus has already passed.)
5. Eu vi-te no centro comercial. (I saw you at the shopping center.)