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

Yes

The word "já" is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The primary difference between the two dialects is not the word itself, but the syntactic construction that typically accompanies it. In Brazil, "já" is frequently used with the gerund to describe an action currently in progress (e.g., já estou saindo). In Portugal, the same idea is expressed using the preposition "a" followed by the infinitive (e.g., já estou a sair).

Brazilian Portuguese Examples

1. Eu já terminei o trabalho. (I already finished the work.)
2. Já estou saindo de casa. (I am already leaving the house.)
3. Você já viu o cachorro? (Have you already seen the dog?)
4. Já faz muito tempo que não te vejo. (It has already been a long time since I've seen you.)
5. Já vou fazer isso agora mesmo. (I am going to do that right now.)

Continental Portuguese Examples

1. Eu já terminei o trabalho. (I already finished the work.)
2. Já estou a sair de casa. (I am already leaving the house.)
3. Já viste o cão? (Have you already seen the dog?)
4. Já faz muito tempo que não te vejo. (It has already been a long time since I've seen you.)
5. Já trato disso agora mesmo. (I'll take care of that right now.)