He does. 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.
Not only was Obi-Wan's memory outright confirmed in The Princess, but small hints also imply the truth even in the original films. Vader did recognize C-3PO, and R2 quite possibly recognized Yoda, but either couldn't or chose not to inform Luke about him.
C-3PO stays behind with Shmi when Anakin leaves for Tatooine with Padmé and Jedi Master Qui-Gon Jinn to begin his Jedi training. The protocol droid later went with Shmi to another home when she went to live with moisture farmer Cliegg Lars.
When the OT was first released, Vader/Anakin had never been a slave boy on Tatooine, had never built C3PO, and never owned R2D2. Hence he didn't have any affect with either in the movies. There are most likely millions of 3PO droids across the Galaxy, we see several during the films. the same with the 2D2 droids.
It was easy to believe that Darth Vader had kind intentions or a simple code of honor for the Sith, but his plans were much more strategic. Instead, Darth Vader wanted to keep Chewbacca alive because he wanted the Rebels to collectively suffer even more. He wanted them to be weak.
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.
After a chase through an asteroid field, the Falcon escapes to Cloud City at Bespin. While exploring a room in Cloud City, C-3PO is blasted by an off-camera stormtrooper.
The base assumption of why Obi-Wan doesn't remember R2-D2 -- in light of the prequels -- is that Obi-Wan is feigning ignorance in A New Hope to downplay the importance of the droid's arrival.
After Skywalker turned to the dark side of the Force, the droid served Senator Bail Organa for a time in the Imperial Senate. Nineteen years following the purge of the Galactic Republic, R2-D2 played a pivotal role in helping the Rebel Alliance destroy the Empire's Death Star superweapon.
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.
Theory 1: Leia Doesn't Remember Ben Is Obi-Wan
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.
Yoda recognized R2-D2 during The Empire Strikes Back.
Vader relies on the complex life support system of his suit to survive and it happens to be very vulnerable to electrical discharges. So, if Vader ever had unleashed Force lightning, there's a good chance it would've caused his suit to short circuit, thus, killing him.
His memory was wiped at the end of Revenge of the Sith. A blabbermouth like him can't be trusted with sensitive information.
C-3PO had his memory wiped at the end of Revenge of the Sith. R2-D2 did not, but why he never told Luke the truth is actually quite simple. That ends up happening when you make the prequels after the originals lol. Bail Organa had Captain Antilles erase C-3PO's mind shortly after the twins were born.
She knows Obi-Wan's real name as well as his alias of “Ben.” When Luke comes barging into her cell and says, “I'm here with Ben Kenobi,” Leia jumps off the bunk in an instant.
In The Phantom Menace, set 32 years before A New Hope, R2-D2 is portrayed as belonging to the Naboo defense forces, one of three astromech droids deployed for repair duty onboard Queen Padmé Amidala's (Natalie Portman) starship as it attempts to get past the Trade Federation blockade.
Yoda claims to have been training Jedi for 800 years. The second time Yoda mentions his long life is in Return of the Jedi, when he tells Luke, “When you reach 900 years old, you will not look as good as you will.” If we take Yoda at his word, that means the Jedi Master died at the ripe old age of 900.
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.
R2-D2 was in on the scheme to keep the secret away from Luke and Leia. He was devoted to the Skywalker family, and at the guidance of Obi-wan and Yoda, he would have maintained that secret in order to help protect the twins.
Anakin was still very young, so he grew up and became a Jedi with R2 at his side. Star Wars: The Clone Wars detailed their friendship at length, devoting entire arcs to it. They were inseparable, even as Anakin's faith in the Jedi Order and the Republic started to falter.
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.