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

Yes

The word "vez" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. It is a noun used to denote an instance, an occurrence, or a turn in a sequence. The only differences between the two dialects regarding this word are found in pronunciation (specifically the phonetic realization of the surrounding vowels and the rhythmic stress of the sentence) and the way it is paired with other words in natural, regional phrasing.

Brazilian Portuguese Examples

1. Eu fui àquela loja uma vez. (I went to that store once.)
2. Desta vez, eu vou conseguir. (This time, I am going to succeed.)
3. Toda vez que eu viajo, eu fico feliz. (Every time I travel, I get happy.)
4. É a sua vez de brincar. (It is your turn to play.)
5. Eu vou à academia uma vez por mês. (I go to the gym once a month.)

Portuguese (Portugal) Examples

1. Eu fui àquela loja uma vez. (I went to that store once.)
2. Desta vez, eu vou conseguir. (This time, I am going to succeed.)
3. Sempre que viajo, fico feliz. (Whenever I travel, I get happy.)
4. É a tua vez de brincar. (It is your turn to play.)
5. Eu vou ao ginásio uma vez por mês. (I go to the gym once a month.)