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

No

The difference lies in typical usage and pronunciation.

While both dialects use "familiar" as an adjective to mean "well-known" or "related to family," they differ when referring to people. In Brazil, "familiar" is rarely used as a noun to refer to a person; Brazilians much more naturally use the word parente to mean "a relative." In Portugal, "familiar" is a very common and standard noun used to mean "a relative" or "a family member." Additionally, the pronunciation differs: Brazilian Portuguese features more open, clear vowels, whereas Continental Portuguese uses more closed or reduced vowels.

Brazilian Portuguese Examples

1. Esse rosto me é muito familiar. (This face is very familiar to me.)
2. Gostamos de ter um ambiente familiar em casa. (We like to have a family-like atmosphere at home.)
3. Ele está passando por alguns problemas familiares. (He is going through some family problems.)
4. Esse cheiro me é muito familiar. (This smell is very familiar to me.)
5. Um parente meu vai viajar amanhã. (A relative of mine is going to travel tomorrow.)

Continental Portuguese Examples

1. Este rosto é-me muito familiar. (This face is very familiar to me.)
2. Temos um ambiente muito familiar aqui. (We have a very family-like atmosphere here.)
3. Ele está a passar por alguns problemas familiares. (He is going through some family problems.)
4. Este cheiro é-me muito familiar. (This smell is very familiar to me.)
5. Um familiar meu vai viajar amanhã. (A relative of mine is going to travel tomorrow.)