That is a profound question, and difficult to give a straight and simple answer. Of course, you have a name and several identities at any given time.
But those are external and ‘perishable’ identities. It is there today and may or may not be there tomorrow. And, hence, it can’t be your permanent and lasting identity. Pandit Shriram Sharma, in his Absolute Law of Karma, says each and everyone on this planet conceptualises an individual according to his own bent of mind. And the man himself is confused all the time in his multi-personal bends and lives with multiple identities while in interaction or relationship with others.
But then the point is that all these multiple identities could help one realise one's multiple roles and hence one's purpose in life too. It helps him become more responsible and ready to face the realities of life. Of course, not all people realise this and, therefore, they pass on without really knowing who they are and what the purpose of their lives is. A precious life goes sheer waste!
To understand the truth of 'Who am I?' one has to go beyond the external identities without giving them up but with them as multiple tasking responsibilities. One has to strive hard to remove the veil of ignorance that one is always shrouded in. Also, we have to get rid of the make-believe world that we are so much used to live in and with. And, remember always, if you don’t know yourself, you will never know and understand any of the people you interact with in life. That is why some people have problems with others, always.
One has to reach a stage where the reality of life is face-to-face and one can't get out of it. Only then, it can be said one has the answer to who am I? The answer does not come easily as it is tagged with the outcome of one's ability to see the reality of life and its purpose. Once you see the light inside, the answer is an automatic outcome.