Sadly, not every person gets a mother who knows how to be a mom. Yours clearly didn’t know how to protect you or love you as she should. Apparently the best she could do was to keep a roof over your head and provide for you. That’s not nothing, but it wasn’t enough. It has left you with an emotional hole you are trying to fill.
But — you have been lucky enough to have alternate mothers. Your grandmother and your friend’s mom can fill the emotional needs your mother can’t handle. In psychology, we call such people “benevolent witnesses.” They are people who provide a positive experience of a loving relationship when a parent can’t or won’t do it. They become the source of loving interest and care in the present and a beacon for how to be a good parent in the future. You have not one, but two such women in your life. This is a gift!
I’m not sure you’ll ever be able to love your mom. You may be able to figure out what made it so hard for her to be a better mother. You may be able to forgive her for not doing better. You may be able to thank her for what she was able to do in spite of her own challenges. Your compassion and appreciation are the kind of love you can give her honestly.
Meanwhile, you don’t have to look far for the kind of motherly love or mother-daughter relationship you seek. You already have them in your grandmother and your friend’s mom.
I think therapy could help you feel okay about the limits of your relationship to your biological mother and would help you embrace the mothering that is available to you.
I wish you well.