SPOILERS FOR THOSE ONLY FOLLOWING THE ANIME.
The fandom jokes about Eri being Aizawa’s daughter because after her arc, it’s hinted that Aizawa’s been looking after her ever since then even though it’s not necessarily his job. We’ve seen him constantly visiting her in the hospital, tying up her hair, and escorting her around. And just recently in an extra page from the new volume it was revealed that he actually went out of his way to buy clothes for her.
UA became responsible for looking after Eri, but Aizawa seems to be the main caretaker. It’s not by any means official, but Aizawa’s spent a considerable amount of time taking care of Eri when he could easily share the responsibility with the other staff. Plus Aizawa’s soft character has only been growing since the beginning of the series and him becoming more of a father figure is becoming a mainstream hope for the fandom LMAO.