we could also ask "Who is born?"
So far the responses in this thread seem to suggest a plausible answer is "not an identifiable, distinguishable self" but instead a what is born. That what being, "human child" or "baby."
So, if we were to ask in another context, when was Albert Einstein born, we might not go with birth of the human child, as that is not distinguishable from all other human children. Albert Einstein is a who, and is referenced in many books (a lot of which are scientific) as a who, and so when trying to be accurate of when was Einstein born, we might have to say sometime later after the human child was born, perhaps about 1 year later is when awareness of the self came into being.
This will help us avoid potential semantical games where 'human child' tells us nothing about 'who is born.'