Originally Answered: Does Darth Vader remember
Further in Empire Strikes Back, neither Yoda nor Darth Vader recognizes either droid.
He either didn't realize it was the same protocol droid or figured he'd been memory-wiped so FUNCTIONALLY wasn't the same droid and didn't care.
Star Wars Characters That Can Understand Droids
While all droids understand Binary, only a handful of living creatures can. Canon characters, such as Anakin Skywalker, whose relationship with R2-D2 is one of the best in the whole series, Rey, Poe Dameron, Hera Snydulla and Kazuda Xiono are fluent in the droid language.
Yoda recognized R2-D2 during The Empire Strikes Back.
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.
R2-D2, the indomitable droid and mainstay of the Star Wars series, may be responsible for one more incredible feat: saving Grogu.
After the battle, R2-D2 and C-3PO were the sole witnesses to the marriage of Anakin Skywalker and Padmé Amidala in Lake Country. After this Padmé gave ownership of R2 to Anakin while she kept 3PO.
Yes…. even in Revenge of the Sith. Artoo was part of a conversation that was very much a secret. While Threepio heard too, his memory banks were wiped clean afterwards.
Yes, he did. In the comic Thank the Maker, Darth Vader comes into possession briefly of 3PO's disassembled head while on Cloud City.
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.
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.
Darth Vader didn't learn he had a daughter until Return of the Jedi, when he stood before Emperor Palpatine. The Emperor's power probed Luke's mind, attempting to goad him to fall to the dark side, and he discovered Leia's existence.
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.
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.
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.
But when Leia admitted she knew her son was gone, Luke told her that “no one's really gone.” As brother and sister touched, Leia understood that Luke wasn't physically there, but a Force projection. His confrontation with Kylo bought the Resistance time to escape Crait, with Poe taking charge.
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.
Fans have been dying for her to meet Luke on-screen for a long time, ever since their knowledge of each other was teased in the Star Wars book The Secrets of the Jedi. Here, it's hinted that Ahsoka inspired Luke to achieve greatness, with Luke himself mentioning that Ahsoka was his father Anakin's Padawan.
Since Obi-Wan and Anakin often fought together, Obi-Wan was friends with R2-D2 as well. This means that although R2-D2 might not have immediately recognized Obi-Wan on Tatooine due to his age, Obi-Wan should have recognized the droid.
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.
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.
As depicted in "Chapter 20: The Foundling," Grogu was rescued from the attack on the Jedi Temple by Jedi Master Kelleran Beq. Beq completed the daring escape that got Grogu to safety, then escaped the attack with his life.
Grogu's force sensitivity is the strongest reason that there is a bounty on him. While force sensitivity is the most important trait that he is being hunted down for, the imperial remnant might be seeking him to study the genetic traits, for cloning, or for training him to become one of them.
With that in mind, it is somewhat surprising Yoda knew it was R2-D2 in The Empire Strikes Back just by looking at him - even though they hadn't seen each other for years. Though R2-D2 is an icon amongst Star Wars fans, astromechs are very common in-universe and many of them don't stand out from the crowd.