Love can be had in any relationship and between any two people. As I've written before, "motherly love" is simply a myth that masks the selfishness of forcing a human into existence that it never asked for. We don't choose our parents and only play lottery in that deal so how can we claim to even "be in love" to begin with? But we do choose friends and people that really give meaning to our life. Perhaps even repairing the damage so often done by the people we call "parents."
Thus the emphasis on stability is just a way for the pronatalist culture to hide its true intention. You don't need to be biologically related to someone for them to be considered family nor does a biological connection mean that you love them. Indeed society could be better off if we would realize that a good, stable home can be in so many areas besides the default biological family. If we truly love and care about someone then we will protect them even from the family they had misfortune of being born into. Love is never rage, never insolent, or never painful. Its only the entitlement of the default that makes it so. We could hand over this poor child of misfortune to the father, and only to be reminded that mothers don't want fathers to have custody and are thus deterred from giving it away. Can we still call this love when its really pawning? Do we deny the souls of many a chance to care far more for a kid? Love should allow you to be open, honest, and patient without fear of being manipulated and controlled. How many could say they ever got this from their biological mother? How can you justify emotional damage when the price was never asked for in this hapless lifetime?