
femaleleopard!
Your post really shows what a caring daughter you are and how frustrating it is when your desire to help gets turned around in unexpected and hurtful ways.
You asked, "can you even take a person with it seriously?" I think part of the answer is yes. You can take their actions and behaviors as serious indicators of mental illness and the emotional pain that a person with undiagnosed or untreated BPD has on an ongoing basis. If your mom has been diagnosed with BPD and she has not undergone intense (meaning for years) therapy for help with her illness, then her coping mechanisms--saying hurtful things to you and abusing OTC medications--are going to be maladpative and make things worse for her and everyone who interacts with her.
You also asked, "Should I allow the things my mother says to me affect me?" This is difficult because while we would like the mean things that people say to us not have an affect, the truth is that mean and cruel words hurt us and this is especially true when a parent says it to a child--no matter how old the parent or the child. I'm sorry that your mom has said and continues to say hurtful things to and about you. Saying that you are not her daughter is a terrible thing for a child of any age to hear, and I'm sorry that you experienced that.
Do you have access to a counselor that you can talk to about the emotional abuse that you have experienced? This is one way to work on dealing with your mom's verbally and emotionally abusive comments.
Another item could be to work on your values-based boundaries. This is hard and takes a lot of practice and willingness to experience the fact that it is not intended to make everybody feel good. It's intended to protect yourself and show compassion for the illness. She probably won't respond well as you begin to establish values-based boundaries, but the point is to take care of yourself--get out of the conversation and you can validate her feelings in the process even if the reasons for her feelings are totally illogical--we all have feelings whether or not they make sense is another issue.

You don't have to validate the invalid--you are not a bad daughter--but you could validate that she feels sad or angry--and then remove yourself to protect yourself. Again, this is so hard to do when you have been conditioned to experience and take abuse. This board is a wonderful community, and I hope there are supportive people in your life who can help you right now, too.
I hope you are doing okay. I'm sorry you are going through this right now. Keep posting!
Pilate