When someone else asks you a question, you are not necessarily impelled or compelled to answer at all, and certainly not to answer it fully and thoughtfully.
Of course, when most people ask a question, they don’t expect, or want, a three-volume exposition on the subject: they want a certain piece of information.
How are you? - generally wants no more than to check that one’s fellow citizen is not in need of assistance at this time. If we ask this of everyone we meet during the day, we collect a sense of the wellness quotient of our community - and impart the same impression to the other persons we’ve met. If three out of the four people we meet answer: “I have a nasty cold.” we know there is a contagious, but non-life-threatening bacterial infection to watch out for. If two of them answer, “Okay, except I couldn’t sleep too well with all the noise from that nightclub on my street,” you know there is situation might have to be addressed. And so on.
Similarly with other formulaic questions regarding work, home life or children: the questioner wants 1. to show an interest in his neighbour and 2. to gain one limited piece of information and 3. to get a measure of the general level of contentment/prosperity/welfare of his community.
There are such formulaic, limited-purpose questions that specifically apply to an office work-place, a school, a hospital ward, a construction project, a train trip, etc.
Nobody, not even Socrates, and certainly not some random stranger - has the right to ask “Who are you?”, except to verify your credentials for admittance to a place, event or conversation. In those cases, all they want, and all they’re entitled to, is a designation: name, rank, function, clearance level.
A psychotherapist might legitimately (and under seal of confidentiality) ask you to consider the question more comprehensively, but can also only deal with part of the answer at any one time. Not your designations: she has that on file. Not your current state of wellness: she can see that in your bearing and manner. What might be most relevant to such an interlocutor in such a situation will be on the order of : Which aspect of your identity has been drawing the most of your attention and energy? And why?
If you wish to contemplate your own identity as a whole… You still can’t. It’s an aggregate of genetic heritance, physical development, talents, proclivities, potential, ambition, experience, memory, relationships, education, thought processes, emotional responses, ideas - a lifetime-thus-far of input and output. By the time you finish thinking about one aspect, it’s already changed from the one you began contemplating. You can integrate ideas and a body of knowledge, but you can’t keep an entire personality under observation at once.