I don't think you read properly. Fl didnt fake her identity. That husband of hers did. She had just transmigrated into the story and was just opened her eyes in the forest. She knew nothing. The husband found her pretty and convenient to use since she had no support and he could fool her easily and he could weave a story that he fell in love at first sight to make his crush jealous. So he quickly bought a fake identity for the FL so he could marry her.
And also so he wouldn't have to suffer any consequences for getting rid of her after he gets his crush.
Is there a novel? Can you share the link please
Is she for real faking her identity? But why though? Not that her husband's family is right, but she messed up too. Who wouldn't be pissed or disappointed when you find out your daughter-in-law faked who she is?