There is no official adoption but yeah since Aizawa is training her hence he is more of a Father Figure. He also trains Shinso so he is his first child. So, you may look at the cute fanfics of Eri and Dadzawa.
Eri was initially in the custody of the Shie Hassaikai, a once-powerful yakuza group looking to regain its former glory. Its leader and Eri's "father," Chisaki Kai or "Overhaul," had been using her blood to manufacture anti-Quirk bullets and even performed inhuman experiments on her.
Most fan works of the Aizawa Family portray Shota and Hizashi being married or at least dating, and having adopted Hitoshi and Eri.
She's the daughter of 'The Boss' – the leader of Shie Hassaikai. He left Overhaul responsible for her care – and no, he wasn't terribly thrilled with what was happening there. Though that had more to do with what Overhaul was trying to accomplish, than what was being done to Eri.
Shota Aizawa
Shota, albeit injured, was brought to the scene so he could erase Eri's out-of-control Quirk. Since then, Shota has been somewhat of a guardian for Eri, sometimes taking charge of her care as she lives with the U.A. faculty in the dorms.
However, in her true nature, Eri holds a murderously hatred towards Shizuku for being Kouki's childhood friend (although she is Hajime's lover) as she wanted to kill her and the rest of their classmates to claim Kouki for herself. She considered Shizuku as one of her mortal enemies (along with Kaori).
Ochaco Uraraka
His relationship with Ochaco does not go beyond that of a professor/student.
A. Eri, one of the main characters in My Hero Academia, is initially introduced as a 6-year-old in her first appearance. In the current timeline, Eri is a 7-year-old child.
Eri's Quirk rewinds a person to a previous state. So, she can theoretically reverse aging. If her power can be applied to herself (this is fairly inconsistent among emitter quirks so it's up for debate), this means she can keep herself young forever.
Eris was closely identified with the war-goddess Enyo. Homer uses the names interchangeably. Her Roman name was Discordia.
Aizawa is married to Hizashi. Aizawa Shota and Yamada Hizashi are married.
IzuMiriEri refers to the ship between Izuku Midoriya, Mirio Togata and Eri.
Did Izuku ever find out that Shinsou is Mr. Aizawa's son? - Quora. Did Izuku ever find out that Shinsou is Mr. Aizawa's son?
Following a scorching defeat at Endeavor's hands, All For One used an unstable version of Eri's Rewind Quirk to revert himself back to prime fighting condition. This drastic measure was irreversible, meaning that All For One would slowly rewind himself into nothingness.
Technically, Chisaki is Eri's uncle, and that's fine enough for most people. However, Chisaki goes out of his way to proclaim that Eri is actually his daughter, which is kinda odd.
Eri ( 壊 え 理 り , Eri?) is the granddaughter of the Shie Hassaikai's boss. She was also the key source of Kai Chisaki's operation to manufacture a Quirk-Destroying Drug. She has been living at the U.A. dormitories since she was rescued.
Eri revealed that she feels like she's so worthless, and she's so depressed, that she has trouble even smiling.
All For One is, quite possibly, the strongest Quirk there is in My Hero Academia. It was formerly wielded by All For One and is currently in Shigaraki Tomura's possession.
Eri's quirk is Rewind, meaning that she can rewind the state of any living thing. This means that she can make someone younger, force them to disappear completely, heal their injuries, or even get rid of their Quirk.
Weaknesses: She has no control over her own Quirk and thus can harm others around her even if they are her allies. She is extremely traumatized by her experiences as Overhaul's prisoner, which can be used to manipulate her.
Eri's Quirk is not only the most powerful, but it has the most potential. She can reverse time to a point before someone exists, which is pretty terrifying, but incredibly useful when done correctly. Old injuries and diseases can be completely removed from a person's body simply by reversing the clock.
Shota Aizawa, also known as the Erasure Hero: Eraser Head or simply Eraser Head, is a major character in the 2014 Japanese superhero manga series My Hero Academia and its 2016 anime television series adaptation of the same name.
This is why Aizawa calls Deku a problem child #animewatching #watchin...
To reiterate, Aizawa did not lose his Quirk completely. However, it is effectively lost as he can no longer use it in battle. Despite being unable to properly use it himself, My Hero Academia Chapter 346 showed that Monoma can copy it and use it effectively, as he did against Shigaraki in the Final War at U.A. grounds.