The question being answered is: “Do R2D2 and C3PO know that Darth Vader is Anakin?” C-3PO? Definitely not. As many other have pointed out, C-3PO had his memory wiped at the end of Episode III, so it is almost certain that he does not know.
Perhaps, Darth Vader just figures that this is another protocol droid and not C-3PO. Obviously, C-3PO does not recognize Vader because his memory was wiped at the end of Revenge of the Sith.
Very few people knew that Darth Vader and Anakin Skywalker were the same person up to his death in 5 ABY. Imperials who knew in the current canon: Sidious (emperor), Grand Moff Tarkin, all or most of the Inquisitors, Grand Admiral Thrawn, various clones in the early empire.
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.
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.
Why Didn't Darth Vader Remember C-3PO in The Empire Strikes Back? Star Wars Explained (Legends)
Did C-3PO know Luke was Anakin's son?
Instead, the diminutive droid appears to have kept his own counsel, perhaps wary of interfering too much in Luke's personal journey. C-3PO lost those memories, but he remained with Han and Leia's family after Return of the Jedi, so can reasonably be assumed to have learned the truth by eavesdropping.
As has been noted in several instances, Artoo has never had a full memory wipe, enabling him to retain his recollections of everything he saw and heard over the decades. This includes his first meeting with Padme Amidala, a nine-year-old Anakin Skywalker, and C-3PO himself during the events of The Phantom Menace.
And while his warnings don't work, at the very least, it shows that Maul became aware of Anakin and what he would soon become. With the rise of the Empire, it's likely Maul caught on to Vader's existence pretty quickly.
Return of the Jedi establishes that Yoda was aware that Anakin Skywalker had fallen to the dark side and become Darth Vader. When questioned by Luke Skywalker, he confided that Anakin was Darth Vader. Later, Obi-Wan Kenobi's Force ghost confirmed that Anakin had indeed fallen to the dark side and become Darth Vader.
Since Bail Organa had C-3PO's memory wiped previously in order to prevent him from spilling the secret of Leia's parentage, it could be that this was a semi-regular occurrence for the droid given he couldn't be trusted to keep quiet about things.
The real reason why Obi-Wan Kenobi, the Lars family, or Darth Vader didn't act in the original Star Wars trilogy as though they knew C-3PO or R2-D2 previously is that the prequel trilogy, in which they first meet the two droids, had not been written yet.
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.
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.
Since Obi-Wan had been in hiding, going by the name Ben Kenobi, he admitted that he was actually Obi-Wan. However, he denied owning R2-D2, acting as if he was not familiar with the droid.
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.
Does KYLO Ren know Darth Vader is his grandfather?
Kylo Ren was always obsessed with his grandfather. He didn't learn he was the grandson of Darth Vader until he was a teenager, when a recording from Bail Organa - intended to be a private message for Leia - was shared with the entire galaxy.
At first, Ahsoka has no idea that Anakin Skywalker and Darth Vader are one and the same. She discovers the truth in Star Wars Rebels when she encounters the Sith Lord in a space battle. What she senses is so shocking to her that she passes out.
