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

Yes

The word "mulher" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation.

In Brazilian Portuguese, the vowels tend to be more "open" and the syllables are more clearly articulated (syllable-timed rhythm). In Continental Portuguese, vowels that are not stressed are often "reduced" or almost swallowed (stress-timed rhythm), and the "l" in "mulher" may sound slightly more velarized (darker) than in many Brazilian dialects.

Brazilian Portuguese Examples

Note: Brazilians frequently use "tinha" to mean "there was" and often omit the definite article before possessive pronouns.

1. Ela é uma mulher muito inteligente. (She is a very intelligent woman.)
2. Minha mulher trabalha no centro. (My wife works downtown.)
3. Eu vi uma mulher no parque. (I saw a woman in the park.)
4. Tinha uma mulher esperando por você. (There was a woman waiting for you.)
5. Toda mulher merece ser feliz. (Every woman deserves to be happy.)

Continental Portuguese Examples

Note: Portuguese people typically use the definite article before possessive pronouns (e.g., "a minha") and use the "a + infinitive" construction instead of the gerund.

1. Ela é uma mulher muito inteligente. (She is a very intelligent woman.)
2. A minha mulher trabalha no centro. (My wife works downtown.)
3. Vi uma mulher no parque. (I saw a woman in the park.)
4. Havia uma mulher à espera de ti. (There was a woman waiting for you.)
5. Todas as mulheres merecem ser felizes. (All women deserve to be happy.)