Although C-3PO is a tell-all, it's possible that he was directly programmed not to say who Leia is. This is something that George Lucas himself suggested in an interview back in the 1970s, which made it into the book The Making of Star Wars: The Definitive Story Behind the Original Film.
Threepio was also there when Luke and Leia Amidala Skywalker were born. They were later left under the care of a Captain Antilles. In order to protect Luke and Leia from their father, Darth Vader, Bail had C-3PO's memory erased, causing Threepio to forget everything from his past life.
C-3PO served Senator Amidala during the Clone Wars who gave him new golden platings and also went on many adventures with his old counterpart R2-D2. Towards the end of the war, Anakin turned to the dark side and Padme died so her friend, Bail Organa took the droid as well as R2 when he took her daughter, Leia.
C3PO was clearly created from some sort of kit or existing prototype since every protocol droid aside from coloring looks the same. He didn't recognize him because again, they all look the same & it was decades until he saw him again. There was no reason to believe he or R2 even existed anymore.
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.
At the end of Star Wars: Episode III - Revenge of the Sith, C-3PO's memory was wiped, allowing him a fresh start by the time A New Hope begins, but R2-D2's memories remained intact, which is why many fans wonder why he didn't tell Luke the truth about his family history, but he might have not done that because he didn' ...
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.
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.
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. "Sister," the Emperor taunted.
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.
We see in legends and Canon that Leia is not the most invested in her jedi heritage. Obi-Wan didn't know there was another Skywalker until Yoda said something. Luke had begun Jedi training and it was easier to focus on one Skywalker at a time. A Jedi can only have one apprentice at a time after all.
The sequence calls back to a similar scene in Revenge of the Sith, when Bail Organa orders Threepio's memory be erased to prevent him from sharing information about Anakin Skywalker's fall to the dark side.
Obi-wan responds to starwars.wikia.com/wiki/Luke_Skywalker saying R2 belongs to him with, “I don't seem to remember ever owning a droid,” which is technically true, because Obi-Wan never did “own” R2 or C-3PO. Obi-Wan then makes a comment that suggests he's more than aware of R2's impressive capabilities.
Maybe R2-D2 didn't know Darth Vader was Luke's father because he didn't see Palpatine christening him Vader, or maybe he believed Anakin was dead because even Obi-Wan didn't know Anakin was alive.
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.
In the year 3 ABY, Fett saved Vader's life from an assassination attempt. In the aftermath of the battle, Vader revealed to Fett his true identity: Anakin Skywalker, and his past with the Jedi Order. It was here where Fett learned of the fate of Mace Windu, and how Vader had killed him.
When Luke Skywalker arrived on Dagobah in 3 ABY, the King of the Dragonsnakes swallowed his astromech droid R2-D2, but then spit the droid back out due to its being indigestible.
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.
