Bipolar is not WHO you are. It's an illness that you have and that needs to be managed. Asking a doctor who you are would be just their opinion on their meetings with you. For me...I wrote down who I think I am - both bad and good. But keep in mind, how you feel impacts this. So I advise that if you do this exercise, do it when you're calm and happy and content.
If I wrote down who I think I am right now (when I am calm but tired and stressed):
Procrastinator
Occasionally lazy
Good with kids and animals
A good listener and can be a good friend
can be selfish especially when I am depressed
Internalizes everything and easily self-blames
Can make others happy, I enjoy making jokes and teasing to make others happy
clumsy
forgetful
a dreamer and a bit naiive
Puts family and God first...and so on.
Once you've written everything down, keep it in your journal or somewhere where you can pull it out. I added mine to my list until I felt like I knew myself better. Then, when I am ready, I'd make a new list.
I'm 28 years old now and comfortable with who I am. Doesn't mean I will stop trying to improve myself, but I know who I am and I am confident in that knowledge.
Something else I just have to say.....I have had a lot of bad experience with terribly doctors concerning my Bipolar but I also have met good ones. Do NOT let doctors determines who you are, do NOT let them tell you how you feel. You know yourself better than anyone. You know your thoughts, pains, side effects, feelings, emotions, etc. Do not let labels like "bipolar" and "personality xxxx" describe you. They're just labels like beautiful, pretty, musician, nerd, gamer.......they are just titles society gives us as a place in the world. You are a person, you are not just bipolar, just moody, just xxxxx......
You are you. Don't think of yourself as "I am personality xx so I have to feel......"
Be who you want to be. Who you are will change as you age and as the world and life affect you. Just remember that you will always know yourself better than anyone else.