Okay, the two LOVE each other.
But we know how it goes in this kind of manga, can't deliver, she is too famous and her family too powerful so running away and incesting with fake identities is unlikely.
Wouldn't surprise me if the author makes that MC existed before his parents got together and no one every told him.
Yes, it doesn't work because the mother would have told the truth, but if the fake will made her think he wanted MC to get in the family...
We will see, this serie have so many twists, I ever expected the fake will truth to be revealed so fast, and to be that.