Is there any concrete evidence that Anakin officially owned R2-D2 or C-3PO?
I found this episode of Star Wars: Clone Wars that shows it, but it's fairly vague...
This is discussed (and answered) in the Star Wars: Revenge of the Sith novelisations. Since it concerns both droids, I'll tackle them in turn:
C-3PO originally belonged to Watto, then Cliegg Lars, then Anakin Skywalker and was finally given over to the exclusive possession of Anakin's wife-to-be, Padmé Naberrie. On her death it's arguable that C-3PO would have reverted to being the property of Anakin/Darth (save for the fact that his death was faked by the Empire) in which case Leia and Luke would have inherited jointly. It's also possible that the two droids would qualify as "sentients" under Naboobian law and would therefore be considered free citizens:
Anakin had only laughed. What use is a protocol droid to a Jedi? Even one as upgraded as 3PO-Anakin had packed his creation with so many extra circuits and subprograms and heuristic algorithms that the droid was practically human.
“I’m not giving him to you,” he’d told her. “He’s not even really mine to give; when I built him, I was a slave, and everything I did belonged to Watto. Cliegg Lars bought him along with my mother; Owen gave him back to me, but I’m a Jedi. I have renounced possessions. I guess that means he’s free now. What I’m really doing is asking you to look after him for me.” “Look after him?”
“Yes. Maybe even give him a job. He’s a little fussy,” he’d admitted, “and maybe I shouldn’t have given him quite so much self-consciousness-he’s a worrier-but he’s very smart, and he might be a real help to a big-time diplomat … like, say, a Senator from Naboo?”
R2-D2 was originally the property of the Naboobian State. He evidently became the sole possession of Padmé Skywalker, née Naberrie after she became Senator and was then passed to the sole possession of Anakin Skywalker. Again, on her death R2-D2 would have become the joint possession of Luke/Leia although there's a good argument to be made that he would have reverted to being the property of the Naboo government:
Padme’s life back when she had been Queen of Naboo, not to mention helping the nine-year-old Anakin destroy the Trade Federation’s Droid Control Ship, breaking the blockade and saving the planet. The Royal Engineers of Naboo’s aftermarket wizardry made their modified R-units the most sought after in the galaxy; he’d tried to protest, but she had silenced him with a soft finger against his lips and a gentle smile and a whisper of “After all what does a politician need with an astromech?” “But I’m a Jedi-“
“That’s why I’m not giving him to you,” she’d said with a smile. “I’m asking you to look after him. He’s not really a gift. He’s a friend.”
Ultimately, with the death of Padme, Bail Organa took possession of both droids (on behalf of his newly adopted daughter Leia) and used them as spies during the Rebellion Era.
C-3PO shuffled along the starship’s hallway beside R2-D2, following Senator Organa who had, by all accounts, inherited them both. “I’m certain I can’t say why she malfunctioned,” he was telling the little astromech. “Organics are so terribly complicated, you know.”
Ahead, the Senator was met by a man whose uniform, C-3PO’s conformation-recognition algorithm informed him, indicated he was a captain in the Royal Alderaan Civil Fleet.
“I’m placing these droids in your care,” the Senator said. “Have them cleaned, polished, and refitted with the best of everything; they will belong to my new daughter.”
R2-D2 is introduced in Queen Amidala’s escape from Naboo. As such, R2-D2 appears to be the state property of Naboo. Queen Amidala as the head of state exercised her control over R2-D2. The Droid continued his service to Amidala while she was a Senator.
Let's consider C-3PO too:
C-3PO was built by Anakin Skywalker. As such, C-3PO is Anakin Skywalker’s personal property. However, as a minor, Anakin’s mother may have held title, but for the fact both Skywalkers were slaves. Moreover, C-3PO was left on Tatooine after Anakin left to become a Jedi. At this point, C-3PO was in the possession and control of Shmii Skywalker as her personal property (assuming a slave could own property).