This is not because the Third Sister knows that Leia is Anakin Skywalker's child; it is because the Third Sister found a link between Obi-Wan and Bail Organa and she believed Obi-Wan would do anything to save Bail's daughter.
Does the Third Sister Know about Luke Skywalker? Apparently, she does. During the episode, Kenobi received a message from Bail Prestor Organa asking about the whereabouts of the Jedi Master and mentioning Luke's location. During the clashes with the Empire, Kenobi lost the transmitter, which Reva promptly retrieved.
It wasn't even that complicated. Third Sister happened to notice, through her studies of Kenobi, that he was close to the Organas. So she hatched the plan to hire a bounty hunter to kidnap Leia, hoping that would draw out Kenobi.
However, Leia's strong-mindedness during Vader's probe denied him the opportunity to discover she was his daughter. That serves to justify Vader's lack of awareness of his connection to Leia, although, perhaps, not as satisfyingly as if Lucas had planned for the relationship from the beginning.
Even though he had a fierce confrontation with Leia Organa during the opening scenes of 'Star Wars Episode IV: A New Hope,' he never even suspected she was his daughter. That was why the only person Vader targeted and tried to convince to turn to the dark side was Luke, as he didn't even know that he had a sister.
That was until Luke Skywalker told her the truth - that he was the son of Darth Vader, and that she was his sister. "It won't be easy for you to hear it, but you must," Luke told Leia before he revealed his secrets to her.
In Bloodlines, Leia stated that she does not forgive Vader, and isn't convinced of his redemption. Surely, she spoke to Luke about how he was able to pull Anakin back to the light. Luke was able to connect with his father before he died, and forgive him for all the bad things he had done. But Leia can't.
However, she chose not to tell Anakin, presumably because he was about to go into a battle. Anakin was prone to letting his emotions affect him, and knowing that Padme was pregnant would've thrown him completely off the task. And Padme probably didn't think that her husband would be fighting battles for months to come.
Leia was facing away from Padmé when she was born as she was being held by a midwife droid. Meanwhile, Luke was in a similar position as he was being held by Obi-Wan. The reason Leia likely remembers Padmé is that she could sense her mother's emotions at the time of the senator's passing.
The Third Sister was at the Jedi Temple during Order 66. She either saw Anakin kill her fellow Jedi or saw the recording of his crimes. That's why she has such hatred for Obi-Wan. She was a Padawan who escaped Order 66.
Theory 1: Leia Doesn't Remember Ben Is Obi-Wan
Throughout her encounters with Kenobi in the series, he refers to himself as Ben. The most straightforward explanation would be that Leia knows the person who rescued her is Ben, and never connected the dots that Ben is actually Obi-Wan.
Seething with ambition, Reva hunts down Jedi survivors of Order 66 as an Inquisitor. She wears the sleek armor of the secret order, and brandishes a unique lightsaber that can split in two to maximize her lethality.
Following the Wampa attack during the movie's opening scenes on Hoth; Leia, Han, and Chewie visit Luke in the infirmary. Leia is annoyed with Han, who insists she has feelings for him, so she kisses Luke to prove she doesn't. Presumably, the scene was simply meant to make Han jealous.
The Inverse analysis — The new revelation that Leia was buddies with Obi-Wan when she was a kid isn't just some cool canon twist, it also adds a tragic layer to everything that happens in A New Hope.
In what exact way is currently unknown. Stuart Beattie's original script for Obi-Wan Kenobi before he was replaced with Joby Harold had Darth Vader killing the Third Sister, who didn't know that he was Anakin Skywalker. She is currently the only Inquisitor to survive a direct battle with Darth Vader.
The answer is seeming, simple: Obi-Wan's self-imposed isolation also includes a news blackout. Not only does he not spread information about himself around, but, he also doesn't appear to take in any new information about what's going on in the galaxy.
Yoda didn't specifically know how this would go down, or to what degree Anakin would cause it to happen, but he was playing a much longer game by Revenge of the Sith, and that's at least, in part, explained by Yoda's Force connection with Qui-Gon's spirit.
Over the next decade, the galaxy was on the verge of civil war with entire star systems—led by Count Dooku, Yoda's one-time Padawan—threatening to secede from the Republic by 22 BBY.
He knows it's Anakin, and the events in this week's episode of The Clone Wars, "A Distant Echo," prove it. And I think Padmé is savvy enough of a political operator to let people think that Clovis could have been the person who fathered the child, if only to avert suspicion from her relationship with Anakin.
Darth Vader only learned he had a daughter in Return of the Jedi - but there's no evidence he realized she was Leia before his death and redemption. Darth Vader may have never known Princess Leia was his daughter in Star Wars and never cultivated a relationship with her, even when he was a Force ghost.
The marriage, witnessed by the droids C-3PO and R2-D2, was kept secret due to Amidala's place as a prominent Senator in the Galactic Republic and because of Skywalker's membership in the Jedi Order; the Jedi Code prevented Jedi from falling in love and forming attachments such as marriage.
Yes, when Darth Vader turned back to Anakin Skywalker and sacrifice himself to save his son, he dies and Obi-Wan uses the force to connect with Anakin's spirit. In that time Anakin apologizes to Obi-Wan, desperately asking him to forgive him for everything he has done to him and everybody over the years.
Leia and Han had three children, and they named their youngest son Anakin after Anakin Skywalker, Leia's biological father. Since Luke was the twin to spend more time with Obi-Wan in the old canon, it made sense that he, not Leia, was the one to honor Obi-Wan when naming his child.
Assuming that the Empire would try to find the child of Anakin Skywalker, Obi-Wan and Yoda decided that Luke was a prime candidate to be a lure for the Empire, and Leia would be the one trained to fight Vader and the Emperor. At the end of Revenge of the Sith, both children were placed into hiding.