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

Yes

The meaning, grammar, and spelling of the verb "ser" are identical in both Brazilian and Continental Portuguese. The differences are strictly phonological. In Brazilian Portuguese, vowels are generally more open and clearly articulated, and the "r" sound varies by region (often being retroflex, like the American English "r"). In Continental Portuguese, the rhythm is more stress-timed, which leads to significant vowel reduction; as a result, the unstressed vowels in "ser" may sound much shorter or almost swallowed compared to the Brazilian pronunciation.

Brazilian Portuguese

1. Eu sou brasileiro. (I am Brazilian.)
2. O cachorro é muito fofo. (The dog is very cute.)
3. Isso é muito legal! (That is very cool!)
4. Ela é minha amiga. (She is my friend.)
5. Nós somos da Bahia. (We are from Bahia.)

Continental Portuguese

1. Eu sou português. (I am Portuguese.)
2. O cão é muito fofo. (The dog is very cute.)
3. Isto é muito fixe! (This is very cool!)
4. Ela é minha amiga. (She is my friend.)
5. Nós somos do Porto. (We are from Porto.)