How does Anthem relate to the search for an identity?
Anthem is one example in a long series of examples of what is usually referred to as "Dystopian Literature"; that is, literature which depicts the antithesis to the usual Utopian concept of a totally free and progressive society which has eradicated all its crime, disorder, etc. and has evolved into a state of social perfection.
Like Brave New World and 1984, Anthem presents Ayn Rand's view of such a society, which appears to be Utopian, but in reality every aspect of the society is corrupt in some way or other. In almost all dystopian stories (with very very few exceptions) society is controlled by some central oligarchy or a network of nodes of control to keep the members of society from deviating from the expected happy bliss of complete compliance with the rules. In Rand's world, a fragment of post-apocalyptic New York City, the people have been thrown back into a medieval world of intense control through primitivistic social structure and governance. The search for identity is the whole point of the novel, bearing out Rand's social theory. However, as Rand points out in her works over and again, man's liberty and individualism is an end in and of itself, so conformity to rules must always be consensual, not forced. Force is the ultimate violation of the individual, so it is the centerpiece of any dystopia. In this world, "We" is the essential collectivistic format of societal structure. Therefore, seeking an identity apart from others amplifies Rand's central notion of Objectivism: Man's existence is an end unto itself, and collectivism is the antithesis of this tendency.
Only when the protagonist evolves enough to see his own individuality is he able to begin to act as an individual, face his own fears, and then defy his society's laws. His eventual liberation is a Rand new dawn of life free from institutionalized collective life.
Ayn Rand's Anthem is based on a society where the concept of an individual identity has been eliminated.
People do not have names and instead are given numbers with which they are identified. All decisions from the time they are born to the moment they die are collectively taken and each person is informed about the same thing at the appropriate time. The extremes that have been reached are evident from the fact that nobody is allowed to have a thought that is different from what the others in the society think. To ensure this uniformity, no one is allowed to be alone.
It is a sin to think words no others think...no transgression blacker than to do or think alone.
The protagonist in the story is unable to accept the terms of the society and sets off on his own path of self identification. He finds himself breaking all rules that have been imposed on everyone to ensure commonness and does things which differentiate him from the others. He has his own preferences, wants to make his own decisions, likes some people more than others and though he himself does not realize this till the end, is in search of a unique identity for himself.
This god, this one word: