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

Yes

The word "falsa" is identical in spelling, grammar, and core meaning in both Brazilian and Continental Portuguese. It is used to describe things that are counterfeit or untrue (such as a banknote or a statement) and people who are insincere or "two-faced." The differences between the two variants are purely phonetic: the pronunciation of the "s" varies (in many Brazilian regions it is a sibilant /s/, whereas in Portugal it is almost always a postalveolar /ʃ/, similar to the "sh" in "shoe"), and the quality of the vowels (open vs. closed) differs.

Brazilian Portuguese Examples

1. Ela é uma pessoa muito falsa. (She is a very insincere person.)
2. Eu descobri que essa nota era falsa. (I discovered that this banknote was counterfeit.)
3. Não acredite nela, ela é falsa. (Don't believe her, she is insincere.)
4. Essa notícia é falsa, não compartilhe. (This news is fake, don't share it.)
5. Ela me deu um sorriso falso. (She gave me a fake smile.)

Continental Portuguese Examples

1. Ela é uma pessoa muito falsa. (She is a very insincere person.)
2. Descobri que esta nota era falsa. (I discovered that this banknote was counterfeit.)
3. Não lhe acredites, ela é falsa. (Don't believe her, she is insincere.)
4. Esta notícia é falsa, não a partilhes. (This news is fake, don't share it.)
5. Ela deu-me um sorriso falso. (She gave me a fake smile.)