Grogu was a Jedi in training in the temple at the time of order 66, so it's entirely possible R2 recognized him.
The theory — TikToker darthjarjar suggests R2-D2 is the one who rescued Grogu from Order 66. It sounds far-fetched, but judging by R2's excited reaction upon seeing Baby Yoda, it's possible they have a history. We also know Grogu doesn't remember who saved him, which explains why the reaction wasn't mutual.
It could be that R2-D2 is excited by Grogu because he is, well, a baby Yoda: R2-D2 won't have seen another of Yoda's kind in a few years, but they did recognize each other on Dagobah, and so it could be that viewers are seeing here.
Sure, R2D2 knew of Yoda, but barely knew or interacted with him personally (as you can note during the prequel trilogy, they have no scenes together). It wasn't like they were friends.
Since Obi-Wan and Anakin often fought together, Obi-Wan was friends with R2-D2 as well. This means that although R2-D2 might not have immediately recognized Obi-Wan on Tatooine due to his age, Obi-Wan should have recognized the droid.
However, Leia's strong-mindedness during Vader's probe denied him the opportunity to discover that she was his daughter. This 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.
Padme gives her droid R2-D2 as a gift, and new co-pilot for Anakin.
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.
Obi-Wan Only Says He Never Owned R2
Note that he says nothing about not recognizing R2; his clever dodge gives the impression that they're strangers, but doesn't actually deny knowing the little astromech. Thus, there really isn't a "contradiction" in the first place—Obi-Wan was just being his cunning self.
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.
In The Phantom Menance, R2-D2 is a droid belonging to the Naboo army, and helps repair Queen Padmé Amidala's starship as it attempts to get past the Trade Federation blockade. Later, R2 becomes part of Qui-Gon Jinn's party in Tatooine and meets C-3PO and Anakin Skywalker.
We know that R2 was on Coruscant at the time of the Jedi purge and temple attack. We know that by the time Anakin arrives back at Padmé Amidala's apartment, R2 is with Anakin, sitting snugly in his starfighter. Surely, given Anakin's proclivity for flying, he flew his starfighter to the temple to lead the attack.
R2 was later able to restore his companion during the Battle of Geonosis. After the battle, R2-D2 and C-3PO were the sole witnesses to the marriage of Anakin Skywalker and Padmé Amidala in Lake Country. After this Padmé gave ownership of R2 to Anakin while she kept 3PO.
R2-D2 is the candidate most suggested when theorizing who could have saved Grogu. R2-D2 was on Coruscant during Order 66. While it did not show the droid during the initial march to the Jedi Temple, he was clearly seen in Anakin Skywalker's starfighter when he visited Padme after the massacre.
Grogu is a Force-sensitive being who was born around 41 years before the Battle of Yavin. He was raised in the Jedi Temple on Coruscant and trained by several Jedi before surviving the end of the Clone Wars and Order 66. Ever since, Grogu was taken into hiding, and was forced to hide his Force abilities to survive.
Grogu appears to be the youngest known member of Yoda's species. He is only 50 years old, and he is still developmentally an infant. For the whole first season of The Mandalorian™, they referred to him only as the Child. It seems that the species is, for whatever reason, quite slow to age.
During the Clone Wars, it was common practice to wipe the minds of astromech droids after every mission to avoid the risk of Separatists taking valued information. But Anakin remained staunchly against this idea for R2-D2, meaning the droid never had to worry about losing the memories of his past.
Because Vader's past life had been obscured by trauma, it's possible that the memories of the droids weren't present at the time. Additionally, most droids are designed to be identical to another of the same model. Vader could have mistaken the two as any other generic droid, not realizing their significance.
The actual answer as to why doesn't Owen remember C-3PO is that it's a plot-hole in the saga, since George Lucas wrote A New Hope with little to no idea of where the droid's long and winding backstory would lead him.
However, R2-D2's memory is not wiped; as a result, R2-D2 is the only surviving character at the end of the saga who knows the entire story of the Skywalker family. Both R2-D2 and C-3PO end up in the possession of Captain Raymus Antilles (Rohan Nichol) onboard the Tantive IV.
And if you're wondering if R2-D2 knows that Luke and Leia are Anakin's kids, the short answer is yes. R2-D2 and C-3P0 were present for their birth, and knew Anakin and Padmé were together.
However, because he saw their dynamic as a source of weakness, he opted to spare them. Now that the Empire had gained the upper hand against the rebels, their survival didn't mean much to him in the grand scope of the galactic conflict at hand.
He hasn't owned a droid because R2-D2 is his friend. No one can "own" him, and to claim to have done so disrespects the countless ways the little astromech has saved the galaxy. If the theory applies, then Obi-Wan cleverly makes note of that while simultaneously hiding his past companionship with R2-D2.
How did Padme die in Star Wars: Episode III - Revenge of the Sith? Originally Answered: How did Padme die? Padme Amidala's official cause of death was that she lost the will to leave, despite the fact that she was perfectly healthy.
In the Jedi Order, emotional attachment and possession were forbidden because it was believed they could lead to jealousy and fear of loss, and ultimately the dark side of the Force; for that reason, Jedi were not allowed to marry.