Since all the years are not the same, some events occur on dates that are not found every year.
The phenomenon can be compared, for example, to a birth on February 29th of a Gregorian leap year.
Even most Israelis write the secular year on their checks and on contracts and celebrate the Gregorian date for birthdays and anniversaries.
A calendrical system is an innovation to help track time and temporally relate events. Thus, if one had started counting from Creation (assumed to be September 7, 3761 BCE), January 1, 2000 was the date 2,103,548, and my birth date would be the easy-to-remember date 2,090,299.
Most calendrical systems, however, have some sort of cyclical nature, usually one that relates to a natural phenomenon.
The actual Hebrew date of his/her birth or the date of going to the mikvah and reemerging as a new Jewish soul? I think the hesitance that many of us have about celebrating our own birthdays (and I've heard that before, as well as experiencing it myself) has to do with our enormous efforts to distance ourselves from who we were pre-conversion. If you don't already know your birthday on the Hebrew calendar, check out the very handy Chabad birthday calculator!
I'd never thought about that before, but that's a very deep question! We aren't supposed to turn our backs on who we were before conversion, nor the people who were important to us.