Anakin Skywalker ignored this procedure in regards to R2-D2, to the chagrin of both the Jedi Council and the Republic Military leadership. The Galactic Empire did not grant its droids any independence and sought to keep them subservient and docile.
A smart, spunky droid who would serve a multitude of masters over his lifetime, R2-D2 was never given a full memory wipe nor did he ever receive new programming, with these factors resulting in an adventurous and independent attitude.
Finding C-3PO devoid of useful information, Bane and his accomplices kidnap R2-D2 and extract the vital data, before memory-wiping the droids of the incident, and letting them on their way.
C-3PO had his memories wiped after Episode III, but R2-D2 did not. However, they were working together on the same ship afterwards, where presumably, they would have conversations.
Summary. Obi-Wan's seeming inability to remember R2-D2 in "A New Hope" was a deliberate choice to downplay the droid's arrival and avoid revealing too much too soon.
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, ...
Given that he has been present in some of the galaxy's most important events and has met characters like Luke, Leia, and Anakin, C-3PO has had his memory wiped a total of three times through the Skywalker saga and The Clone Wars, but why?
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.
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.
C-3PO's wipe has been the core of many continuity discussions in the Star Wars trilogies, as he didn't recognize Obi-Wan Kenobi in A New Hope but had known him since Kenobi was a Padawan. By wiping his memory, it allowed the leeway necessary to bridge the logic gap created by the previous two prequel films.
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.
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.
In Star Wars, younglings were traditionally recruited into the Jedi Order from an early age — so young that 9-year-old Anakin Skywalker faced opposition as being too old to train. However, during the original trilogy, Yoda and Obi-Wan Kenobi resolved to train a 19-year-old Luke Skywalker.
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.
Unfortunately, Rex and Vader never meet again though. Why didn't Palpatine feel afraid of showing his evil actions, behaviors and ways to Anakin and the Jedi anymore?
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.
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.
