Are there any cultures or societies where a person's date of birth is NOT the day he or she came out of the womb? ~

The Chinese count from conception.

Sure.
In traditional Aboriginal cultures in remote Australia, there is no date because the languages contain no numbers after three. Most have only one, two or many.
Annual cycles are recognised via the patterns of the stars - extremely sophisticated knowledge used for navigation - and the flowering and fruiting cycles of plants.
Time itself has a different meaning, with events being woven into the Dreaming without chronological order.
The person's birthday is marked by where the mother was at the moment when the fetus first kicked. Whatever spirit occupies that place in the Dreaming is the baby's spiritual father. That place in the landscape will be sacred to the baby, that spirit will be especially sacred, and the animal or plant representatives of that spirit will be taboo to eat. This post was edited by Benedict Arnold at January 2, 2018 6:57 PM MST

Any culture that does not count time.

We got any of those?  

Yes, there are people right here in the US of A who actually believe that. 

not that i know of