The Treasure

Humans and dragons live along side each other, though it isn't exactly peaceful. Dragons have a tendancy to attack human settlements for tressure. Well, had. As a solution, humans started sacrificing their own to the dragons, as a peace offering. These sacrifices were known as 'Treasures'. Ever since, the dragons stopped attacking. Well, for the most part, anyways.

All of that was before quirks started showing up. Once humans started to develop quirks, though, they sacrificed others less and less, instead choosing to fight back. Though, some places continue the sacrifices, even today.

Shouta Aizawa liked to think that he was a rational dragon. All of his choices had a reason behind them, typically one based on logic. For example, he chose to digsuise himself as a human because it made getting shinies easier than stealing. He chose to live in a cave instead of with the humans because a cave didn't cost anything. He chose to become an underground hero because his abilities were more suited for it than above ground heroics, and it made it less likely for him to be discovered.

When he found a small child that smelled like a Treasure out in the forest during the winter, his choice wasn't based simply on logic, but also on emotion. He couldn't leave a kid out there to die of hypothermia, so he brought the kid to his cave. Then he got attached.