I will argue that identity is one thing every person has. Even if one does not have a name, his identity can be considered as nameless.
If people have a choice to choose all the identities they want, they will definitely make full use of it. As we look at ourselves in the mirror everyday, we think of how we want to portray ourselves, how people should see us and hence identify us. Isn’t that how we struggle to choose our hair-cut, our clothes or even the way we speak? It is a matter of self-promotion.
However, not everything turns out the way we want it to because of different perceptions. Personally, I have seen some different ways people identify because of perception. When I wore a pink shirt to work, my male colleague spent the whole day identifying me a as a gay but my female colleague walked up to me and told me that I exuberated confidence just by simply wearing that shirt. So in one day, I was identified as a gay and a confident person.
These identities can be changed over time but identities based on biology will not change as easily. If you are born a black, you will never be a white. If you are born an Arab, you can never change that fact. In recent times we have seen how Arabs are unjustifiably considered to be terrorist. An identity that they do not like but face trouble shaking off.
What I am trying to point out is that even if we try to project our own identity, it is never up to us to decide if it will really be.