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.
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.
C-3PO had been given new parts and a new color. C-3PO himself could not jog anyone's memories as his mind was wiped. Had anyone recognized him, he would have assured them they had never met before. He was also one of a very common protocol droid line; we literally are led to under that there are millions like him.
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.
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, ...
Why Didn't Darth Vader Remember C-3PO On Cloud City?
Does R2-D2 know that Vader is Anakin?
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.
According to Imperial Estimates in Star Wars Inside #87, only about 100 of the Jedi survived Order 66. Most were subsequently hunted down and killed in the years following the Great Jedi Purge.
Further in Empire Strikes Back, neither Yoda nor Darth Vader recognizes either droid. Luke travels with R2-D2 to the Dagobah where they find Yoda. Yoda's relationship with R2-D2 is not on the same level with the same characters. Perhaps, the old creature simply forgot.
They aren't abbreviations — they're model and serial numbers. R2 is a second generation R-series astromech, serial number D2. * And C-3PO is a C-series protocol droid with the serial number 3PO. *And likely several more numbers that get dropped for the sake of time.
The lack of recognition could have come from an old man simply not remembering the passing droids of his youth. Lastly, Owen didn't seem the sentimental type to form any emotional attachments to droids. Being a practical man, he viewed them as tools to be used and replaced when they no longer functioned.
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.
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.
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.
Sound editor Walter Murch states that he is responsible for the utterance which sparked the name for the droid. Murch asked for Reel 2, Dialog Track 2, in the abbreviated form "R-2-D-2".
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.
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.
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.
