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

Yes

The differences between the use of "sensível" in Brazilian Portuguese and Continental Portuguese are purely phonetic. There are no differences in spelling, grammatical function, or core meaning.

In Brazilian Portuguese, vowels are generally more open and clearly articulated, and the rhythm is more syllable-timed. The "e" in the first syllable is clearly pronounced. In Continental Portuguese, unstressed vowels undergo significant reduction or "weakening." Consequently, a Portuguese person might pronounce the word with a much more closed or almost elided initial "e," making the word sound closer to s'sensível.

Brazilian Portuguese Examples

1. Ela é muito sensível a críticas. (She is very sensitive to criticism.)
2. Minha pele é sensível ao sol. (My skin is sensitive to the sun.)
3. Este é um assunto muito sensível. (This is a very sensitive subject.)
4. Ele é um rapaz muito sensível. (He is a very sensitive boy.)
5. O sensor é sensível ao movimento. (The sensor is sensitive to movement.)

Continental Portuguese Examples

1. Ela é bastante sensível a críticas. (She is quite sensitive to criticism.)
2. A minha pele é sensível ao sol. (My skin is sensitive to the sun.)
3. Este é um tema bastante sensível. (This is a quite sensitive topic.)
4. Ele é um rapaz muito sensível. (He is a very sensitive boy.)
5. O sensor é sensível ao movimento. (The sensor is sensitive to movement.)