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.
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. It's like if you sold your car and saw it 20 years later with a different pain job; you wouldn't recognize it.
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.
The protocol droid later went with Shmi to another home when she went to live with moisture farmer Cliegg Lars. Anakin reunites with C-3PO only after tragedy strikes and he returns to Tatooine too late to find his mother kidnapped and killed by Tusken Raiders.
Senator Organa's ship, Tantive IV, is commanded by Captain Antilles. Bail's final line of the film and instruction to the Captain, says “…have the Protocol droid's mind wiped.” Because Bail Organa had them wipe his memory at the end of Revenge Of The Sith and he never came back to Tatooine until A New Hope.
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.
In the comic Star Wars Tales #6, we see an interaction between Vader and C-3PO. In the comic, Boba Fett brings Vader C-3PO's remains. Vader instantly recognizes C-3PO, and is reminded of his childhood.
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.
Vader himself saves Threepio from destruction in the final pages of Marvel's Star Wars, before the series gets a relaunch taking place after Empire Strikes Back.
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.
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.
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.
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.
According to official tie-ins, Anakin's Force Ghost would visit Luke many times over the next decade, offering him guidance and counsel as he reestablished the Jedi Order.
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.
