There are people whose birth date is not really known (babies left at doorsteps, people from war-torn countries with no official record who had no parents to tell them of their birth date...).

I do not know how are they handled, but they probably get assigned one plausible date and it probably depends on the country.

In a sense, a birth date can be just as much an assigned number as an ID is. An ID can also have a checksum in it, potentially even a cryptographic one that only government can sign.