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.
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.
Darth Vader acts like he doesn't even know C-3PO or R2D2. They were with him since he was a child. He did everything he could to forget his past. Also, his plating had been replaced many times, and protocol droids all look very similar.
R2-D2 witnessed some of Anakin's best moments and knew him when he was an innocent boy on Tatooine; it might have been healing for Luke to hear these stories. Perhaps, like the stories about Padmé, they were simply too painful to share.
in the Dark Horse comic series star Wars Tales, which is now part of legends, it's revealed that Darth Vader did recognize C. 3PO on Cloud City. when 3PO was shot by Stormtroopers, his parts were brought before Vader. because the Imperials thought he could hold valuable.
As far as you know, Anakin Skywalker and Captain Rex were very close friends during the Clone Wars. But before the battle for Coruscant, they had to split up and Rex went with Ahsoka to Mandalore, and Anakin went to Coruscant. After that, Anakin and Rex never saw each other again.
Why didn't Maul tell Ahsoka that Palpatine was Sidious?
Maul Was Terrified of Palpatine
Maul had killed Qui-Gon and numerous other Jedi and started a huge Mandalorian conflict that left thousands dead. If he publicly announced that Palpatine was an evil Sith Lord, there would be few (if any) who would take him seriously.
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.
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. And after spending decades with R2 himself, Luke Skywalker is also able to understand him.
Darth Vader's true identity as Anakin Skywalker was a highly guarded secret, known only to a select few individuals including Palpatine, Obi-Wan Kenobi, Luke, Han, Leia, R2-D2, C-3PO, Master Yoda, Bail Organa, Ahsoka Tano, Darth Maul, Grand Moff Tarkin, Grand Admiral Thrawn, Jocasta Nu, Ferren Barr, Verla, Reva, ...
After decades of missions with R2-D2, Luke Skywalker was reasonably fluent in droidspeak. The most common way for binary droids to communicate with organics was by writing, conveyed via another computer system. This is how astromech droid copilots would speak with the pilot of a starfighter.
She senses Anakin when she reaches out to the pilot — and Vader realizes that Ahsoka, his former Padawan, lives on. In Season 2, Episode 18, “Shroud of Darkness,” Ahsoka's guilt manifests as a vision of Anakin, who's angry with her for leaving him all those years ago.
secondly, in Legends, R2 was given programming that actually restricted him. from sharing any information with Luke or Leia. about their parents or relation to one another. this was all part of the plan. to conceal this information from the twins.
Leia's Force abilities and her lack of awareness of her true identity also contributed to Vader's inability to sense their familial connection. Vader's denial of Anakin Skywalker's legacy and his fear of getting hurt again prevented him from realizing the truth about his children until later in the original trilogy.
Three months after the Battle of Endor, Skywalker used the lightsaber during a mission to an Imperial research base on the planet Vetine in order to retrieve the last remaining fragment of a Force-sensitive tree. Skywalker gave his lightsaber to R2-D2 for hiding as they entered the Imperial base.
By wiping his memory, it allowed the leeway necessary to bridge the logic gap created by the previous two prequel films. However, the same could not be said for R2-D2, who never had his memories erased.
