You're not wrong. But I think that statement is supposed to illustrate the parent's attachment to their child despite the horror show going on in their head.
It depends on how you think about it. I think of it as giving my kids a piece of myself in my love and effort to care for them. They are their own people, but they carry a part of me, just as I carry a part of my parents and those who helped shape me into the person I am.