I will do my best to answer your questions but please remember this is just my experience and my opinion from my experience.
With regards to the medication and your mother; medication does not make somebody who they are, at all. I have been medicated for anxiety and depression for almost 14 years now. My beliefs, my morals, the way I treat people, my passions, my athletic ability, etc....they all still exists and are very much who I am. The medication does not change who I am at all. What it does is help to control the symptoms of an illness. Depression and anxiety are not who I am either, but both cause symptoms that are difficult to deal with. The medication simply deals with those symptoms and helps me to not be debilitated by an illness. If your Mother were to stop taking her meds, she may find she is ok. Some people are able to use meds to get things under control, and come off them and live successfully without relapse of full blown symptoms. Others, like me and maybe your mother, need the medication to keep the chemical balance in our brains level so that we don't face the symptoms so heavily. I wouldn't look at it as her meds make her the person she is, but instead, the meds allow her to be the person she really is.
Now as for the hereditary possibilities of facing mental health issues yourself; There are ties to genetics and mental health, similar to a lot of illnesses. It is possible that this is something that may effect you in your life, it is also possible it may not. There are many factors that can lend to Mental Illness and many of which you can't control. It's not a choice to be sick or not. I can say with honesty I don't know what "caused" this for me. There is Mental Illness in my family, I have had pressures and stresses in my life that some may consider extreme, I could come up with a thousand possible "triggers" or "whys" but none of them matter and I couldn't have avoided any of them. What I suggest is first, don't focus on the possibility you may get sick someday. That possibility always exists focusing on the negative will only increase the likelihood that you will experience it. Find comfort in the knowledge that you are surrounded by a family that understands and should you be faced with issues, will be able to support and provide guidance to help you. Pay attention, you have seen what anxiety and depression look like, if you feel you may be experiencing issues, go seek help. Think of it this way, you are armed with a lot of experience and knowledge and should you face any problems yourself, you will be in a far better position yourself to identify it, seek support and learn to cope with it.
Most importantly again however, do not assume it will happen to you, instead just take good care of yourself and do the things that keep you healthy everyday.
Hope this helps,