Indirectly she is the daughter of the mage ( emperor's brother) and officially dukes daughter because she was born or to say made when the mage had possessed the Duke . The mage unknowingly got involved with the fls mother while the Duke was already spending time with the fls mother now I don't know how to say this in anymore angelic words but they did it while the mage's soul with his powers and the dukes soul came in one body. Thus mage's powers seeped in the fl's body. this is how she is an illegitimate child or in simple words as per the Oracle has the blood of the royal.
As I said before the mage reside's in the duke's body but just does not have control over the body unless it is full moon night or at night.
Same way the fact that mage is possessing/residing in the dukes body is realized by the fl's mother on the same day.
About if the fl has both the mage's and duke's power is not conformed by me so i don't know but for now it is known that she has the powers of the mage. There is a 80 % chance that she has both as in the story mentioned people who have Compatibility with mana and magic are rare and on top of that it is always mentioned about how strong she is for a child which could be result of the duke's mana. So you can say that she has both powers.
I don't think people are understanding the plot of this manhwa this isn't just another manhwa i have seen till now there is no illegitimate child setting or abused abandoned child stuff
Spoiler alert
In short
Fl - reincarnated form another world as baby. remembered her memories when 3 years old. Read a online novel or manhwa. Destined to die at age 19.is an extra .Had a past life almost like a protagonist of a betrayed child novel. Not illegitimate. Doesn't have any plans like I won't be begging for ur love / I will leave as I please / etc etc.
World of reincarnated - seem to be real and just like any other world with laws and all.
Og Fl - is as well a reincarnated but her past life is different from what was mentioned in the novel fl had read. Wants fl's strong mana might have a back story related to og fl's past life about the mana . Daughter of the empress.
(The emperor of this new world has a brother who might be a mage)
Fl 's father in this new life -
3 years before fl's birth this emperor's brother found himself possessing fl' father's body. Sitting in front of him was the naked fl's mother. That's how fl is indirectly illegitimate child. This guy's mana got liked and absorbed in fl's body as in fetus I guess. This mana dude can only have control over fl's father's body on full moon nights. So basically his body has two souls residing in one his own second this mana dude.
Fl's mother- is a duke or a general.never betrayed or cheated on fl's dad. After fl's birth went on some journey for some reason. Died because of ship sinking in ocean because of some pirates.Is the best friend of the empress.
Og fl's mother AKA empress- og fl's mother. Went with the fl's mother on ship. Died of ship sinking same reason. Didn't know she was pregnant with og fl. Her head with baby og fl was sent back in a box.
Emperor - known as heartless tyrant . Told his people to confine the og fl baby who ever would go against this order will be beheaded. This was his way to get over empress death.