By claiming he's never owned a droid, he applies the same tactic as he did with his embellishment about Anakin. He hasn't owned a droid because R2-D2 is his friend. No one can "own" him, and to claim to have done so disrespects the countless ways the little astromech has saved the galaxy.
And that's just one of many reasons fans of the franchise scratch their heads when he reunites with Jedi Master Obi-Wan Kenobi in A New Hope. Despite years of adventures together, the Jedi doesn't appear to recognize the droid at all.
However, R2-D2's memory is not wiped; as a result, R2-D2 is the only surviving character at the end of the saga who knows the entire story of the Skywalker family. Both R2-D2 and C-3PO end up in the possession of Captain Raymus Antilles (Rohan Nichol) onboard the Tantive IV.
Meanwhile, R2-D2's cameo in Obi-Wan Kenobi comes a bit later on at the 38:45 mark of the episode. He's briefly shown on the right side of the screen as Bail and Breha Organa talk about Leia. The appearances of R2-D2 and C-3PO in Obi-Wan Kenobi work perfectly within the established Star Wars canon.
Obi Wan and R2 interact frequently in the prequels - given the timeline, it seems odd that Obi Wan would have absolutely no recollection of R2. Unlike the droids (i.e., C-3PO and R2-D2) Obi Wan's memory wasn't wiped.
R2-D2 was present when Padmé gave birth to Leia and Luke, so he knew Luke was Anakin's son – but what he didn't know was that Anakin became Darth Vader. In addition to that, Anakin's fall was surely a painful event in R2-D2's life, one that he definitely wouldn't like to remember or even talk about.
Further in Empire Strikes Back, neither Yoda nor Darth Vader recognizes either droid. Luke travels with R2-D2 to the Dagobah where they find Yoda. Yoda's relationship with R2-D2 is not on the same level with the same characters. Perhaps, the old creature simply forgot.
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.
The real answer is: When the original trilogy was written and filmed, Anakin Skywalker was not the creator of C3PO. There was no consideration given to who Vader would recognize, because there was no reason for Vader to recognize the droid, at all.
Obi-Wan Was Pretending He Didn't Know R2-D2
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.
C-3PO, his robot companion, could speak 6-million languages including excellent English with a hint of a British accent and all R2-D2 could muster were beeps and whistles. Clearly the technology was there to let him speak, but whoever designed him intentionally decided to stick with the beeps and whistles.
As such, R2D2 appears to be the state property of Naboo. Queen Amidala as the head of state exercised her control over R2D2. The Droid continued his service to Amidala while she was a Senator. C3PO was built by Anakin Skywalker.
Sure, R2D2 knew of Yoda, but barely knew or interacted with him personally (as you can note during the prequel trilogy, they have no scenes together). It wasn't like they were friends. So Yoda shows up and he is acting crazy and stealing food. R2 doesn't care that this guy is a Jedi Master, he's not standing for that!
There is no evidence that he remembered Obi-Wan. He may have been looking for Obi-Wan because Leia added it to his programming while loading the message, but didn't have any real memories of Obi-Wan from before.
Does Darth Vader remember being Anakin? Yes. Yes, he remembers his name but feels it has no meaning to him anymore.
As we all know well, in A New Hope, Obi-Wan went to Mos Eisley to meet with Chewbacca and Han, in order to get off Tatooine to get R2-D2 to the Rebellion. By the end of Revenge of the Sith, Obi-Wan and Chewie didn't meet, at least that we have seen.
Darth Vader can not use force lighting, because of his severed arms. Robotic hands can't summon the force lighting. However Darth Vader can still use other forms of the force, such as force choke.
Vader agreed to help the handmaiden, silently recalling the time he spent with Padmé in the process. In this moment, he was once again Anakin Skywalker, a young man in love who simply wanted to change the world for the better.
C-3PO accidentally became part of the Battle of Geonosis when his head was affixed on top of a battle droid's body after a mishap in the Geonosian droid factory. R2-D2 and C-3PO witnessed the secret wedding between Anakin Skywalker and Padmé Amidala on Naboo.
Yes. In the book 'Tatooine Ghost', Leia learns of Anakin's life prior to his evolution into the Darth Vader personae. This revelation leads her to forgive him. She even named her third child in memory of her father.
It's hard to say for certain - but even if Vader was aware of Yoda's existence, chances are he probably did not know where Yoda was; being the Jedi Master that he was, he had an exceptionally strong connection to the force - and so, he could have easily hidden his location and energy in the force whilst on Dagobah, all ...
One of the strongest and most compelling reasons that Vader senses Luke on his approach to Endor, but the Emperor does not, is that Vader is both Luke's father and conflicted himself.
Senator Bail Organa takes ownership of R2-D2 after Anakin completes his metamorphosis into a Sith Lord. By proxy, he is the enemy of the Empire, as Organa is a staunch supporter of the Rebellion.
Anakin Skywalker is the son of Shmi Skywalker, born without a father through the Force. He is the secret husband of Padmé Amidala, the father of Luke Skywalker and Leia Organa, the father-in-law of Han Solo, and the maternal grandfather of Ben Solo.
R2 doesn't know it. It's as simple as that. He isn't present on Mustafar (he flies with Anakin but is ordered to stay at the ship. He isn't present while the slaughtering of the seperatists or the fight of Anakin/Obi-Wan.), differently to Obi-Wan and C-3PO.