So it is kind of explained in Chapter 4. The previous emperor went mad after Eclette's mom(a saintess) left him. The younger brother of that emperor assumed the throne because of the previous emperor's madness. Our MC is in the body of the current emperor's daughter.
The whole false princess thing is a weird way of putting it. There is nothing to suggest that she isn't a real princess, she is the current emperor's daughter. Her twin was made the crown prince. She doesn't, however, have saintess and royal blood like Eclette.
The MC mentions that the empire was founded by a woman but since then all of the rulers have been men. I think that may be because until Eclette there hasn't been a princess with saintess blood. Or with both red hair and gold eyes . All like the original empress.
I think that Gibrilette person is just a zelot. So the brand "fake princess" has more to do with her and the religion in this story's feelings about a princess with saintess powers than the MC's bloodline.
Seeing that the MC's fiancé is stuck up, he was probably only marring her for power. And he switched to Eclette since she was most likely to become the ruler.

So she's really not the current emperor's daughter since she said that she's not his daughter by blood in chapter 10.
So her being a fake princess, meaning that she's a child of her mother from another man before her mom goes into the palace and becomes the wife of the current emperor? Is that how it is?