In the film, it was established that Leia once trained under her brother Luke to become a Jedi. The training even went far enough that Leia crafted her own lightsaber, though eventually, she decided to abandon the path of the Jedi and become involved in New Republic politics.
Still, it's a shame that Leia's training never took the next step. Her kind-heartedness would've been perfect for a Jedi of the new Order. However, it wouldn't be fair to say her potential got wasted, as she was able to train Rey and used her strength to help turn Ben Solo back to the light side.
After the Battle of Endor, Leia trained as a Jedi with Luke, but left the Jedi path after sensing it would result in the death of her son. She returned to the political arena, where the New Republic had reached an uneasy peace with the remnants of the Galactic Empire.
While Leia wielded the Force in order to sense Luke throughout the later part of the original trilogy, she wasn't seen using the Force in a grander and more tangible way until she saved herself from the vacuum of space in Star Wars: The Last Jedi.
In 18 ABY, her twin brother Luke Skywalker gifted her a lightsaber that contained a ruby-red colored blade. Until Star Wars: The Rise of Skywalker, Star Wars geeks were unaware that Princess Leia wielded a lightsaber.
Leia's Anger Would've Made Her Strong In The Dark Side
Leia's capacity for anger is so great even without her corruption to the dark side that she remains reluctant to forgive her father for his actions as Darth Vader, even after his redemption and despite Luke's willingness to forgive.
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.
Leia was not using the Force or was currently Force-sensitive during her confrontation with Vader. She began training as a Jedi years later. All these reasons prove my point: Vader didn't know Leia was his daughter, and felt no presence of the Force in her, as she was not currently strong with the Force.
Summary. Darth Vader's failure to recognize Leia as his daughter in Star Wars: A New Hope stems from George Lucas's late decision to make Luke and Leia siblings. Leia's Force abilities and her lack of awareness of her true identity also contributed to Vader's inability to sense their familial connection.
Lucas addressed this. If Princess Leia had become a Jedi, could she have been more powerful than Luke? Speaking for Legends, she does. In Ambush at Corellia, Luke and Leia have a duel after he gives her a new lightsaber and she actually defeats him.
Yoda may have wanted to train Leia instead of Luke, but he was wrong - precisely because he did not understand that the redemptive love of a son towards his father was what would bring balance to the Force. Fortunately, the will of the Force won out, and Luke was the one chosen to train as a Jedi instead of his sister.
Due to her powerful Force heritage, Leia, following her family's legacy, became a Jedi Knight in the New Jedi Order. She was partially trained by her brother Luke and later by Jedi Master Saba Sebatyne, who declared her a fully trained Knight at the end of the Swarm War.
As Leia did not discover that she was a child of a Jedi until ROTJ, she would not have been able to intentionally use it. She also did not have any training so her use of the Force would be limited compared to other people who had been trained and studied it for years.
Princess of Alderaan. The royal couple of Alderaan adopted the late Padmé Amidala's infant daughter, renaming her Leia Organa. Through adoption, Leia Amidala Skywalker became Leia Organa, formally becoming heiress of the Royal Family of Alderaan through the Name Day ceremony.
Over one year after the battle of Endor, Anakin appeared as a forest ghost in front of Leia. to ask her for forgiveness. When Leia saw him, she was horrified. and could not find it in her to accept him. She remembered all the people he slaughtered, what happened to Alderon, and the thing that bothered her the most.
Vader was hunting down the pilot who blew up the Death Star, and hired Boba Fett to find them. Boba was unable to, but he did give Vader a name: “Skywalker.” This, combined with his knowing Luke was Force-sensitive and had been trained by Obi-Wan, was what made him realize the truth.
The twins are quickly hidden from Anakin and each other in order to protect them. Years later, Darth Vader is astute enough in the Force that he not only senses Obi-Wan's presence in A New Hope but also identifies Luke Skywalker as his son in The Empire Strikes Back's earth-shattering conclusion.
In the post-2014 canon universe, the story that reveals Darth Vader discovering Luke Skywalker's identity (and thus the fact that the Death Star's destroyer is his son) is issue 6 of 2015's Star Wars comics by Marvel, which takes place sometime shortly after A New Hope.
He knew Luke was, but not Leia at first. In the first two Star Wars films, Vader comes face to face with her multiple times and doesn't recognize her as his daughter. As for Luke, he knew from the moment he heard the name he had a son and tried to find him.
Bail Organa was present during the climactic end of Star Wars: Episode III - Revenge of the Sith, and there can be no doubt he was aware of Darth Vader's true identity.
And during the conversation, Padme is holding her stomach and appears to have a bump, seemingly confirming she was pregnant long before Revenge of the Sith. However, she chose not to tell Anakin, presumably because he was about to go into a battle.
The base assumption of why Obi-Wan doesn't remember R2-D2 -- in light of the prequels -- is that Obi-Wan is feigning ignorance in A New Hope to downplay the importance of the droid's arrival.
