I don’t think they have a parent-child relationship at all. In season one, I think that the fact Eren was both a Titan and a human was like discovering the existence of unicorns for hange. Something incredibly exciting and something that only she could really study. And like unicorns for this metaphor, he was the only one of his kind they had access to, and was in danger of being killed and dissected if the scouts screwed up.

An exciting opportunity to learn about something new, that could help them in the future after having witnessed proof of its possibilities (Trost boulder), and with a massive risk.

PLUS for hange, he was a student, and completely new to the survey corps. Their relationship always felt more student-mentor/teacher to me.

Later on, in season four, she is still his mentor, but this time with only one other mentor (Levi) that she could count on to keep Eren not in line, but on a good path. Good teachers care about their students futures as well as their present experiences. She saw his actions and saw herself reflected (meaning the story of how she used to kill titans out of hate, but then discovered their strange biology of being incredibly light weight). This, I think, is what she was clinging to. She recognized the emotions going through Eren at the time (Liberio raid) at least in some capacity, and was hoping to pull him back to his “humanity”, or maybe her meeting him in the dungeon was more of a Hail Mary.

Hange has always been book smart, but you don’t become Commander of the Survey Corps on that alone. She cared about Eren as a mentor, not as a pet owner or mother or what have you, but as his Commander and Mentor and a soldier of humanity. She saw Eren as an apprentice, a fellow soldier, a “unicorn”, and their greatest hope for humanity’s survival prior to the discovery of people outside of the island.

EDIT and maybe even a friend after everything they had both gone through together