It's homophobia to an extent. Miok came from a family where she did not recieve love. Her parents live away from her and dont care at all for her. That makes her idolize the "normal"/traditional family unit where there's a mother and father who live together and take care of their child. She is sensitive to any form of "abnormality" because for her it represents emotional neglect from the parents. In this case , the "abnormality" was a family unit with two dad's because in her head, it's abnormal and it won't Lead to a happy family.
She is an emotionally neglected child who even after growing up hasn't been able to fully deal with the trauma of having emotionally neglecting parents. This came out as an irrational actions as she fully believed that she was doing something good by forcing him to date a woman. It's sad.
I'm rereading the whole thing now and I wonder. Is the reason why Miok "can't support them" not because of them but because of the little girl? I kept getting confused why she was being so weird about it and that would be the only somewhat reasonable explanation. That she doesn't want the girl to feel singled out once she's older, just like Miok feels now. If not, then the whole arc doesn't make much sense to me.