Why was Frederick Douglass not sure when he was born?

Frederick Douglass did not know when he was born because slaves were not told their birth dates.
One of the differences between black children and white children was that white children knew their birth dates and black children did not know theirs.  It was a small measure of inequality that was symbolically meaningful.  It indicated that black children were not valuable enough to know the dates they were born or exactly how old they were. 

I have no accurate knowledge of my age, never having seen any authentic record containing it. By far the larger part of the slaves know as little of their ages as horses know of theirs, and it is the wish of most masters within my knowledge to keep their slaves thus ignorant. (Ch. 1) 

Douglass comments that he never really knew a slave who knew his or her birthday.  They could only tell the time of year in which they were born.  Yet, as a child he wanted to know his birthday, because the white children knew theirs. 

A want of information concerning my own was a source of unhappiness to me even during childhood. The white children could tell their ages. I could not tell why I ought to be deprived of the same privilege. (Ch. 1) 

Even to ask his master his birthday would have been considered inappropriate.  This was “evidence of a restless spirit.”  It annoyed Douglass not to know his exact age.  He was a person, not an animal.  Not telling the slaves their birthdays was another way of subjugating them and keeping them down.
Slaves were not really able to keep accurate records of birth dates among themselves, and it probably was not a priority for them.  They were more interested in surviving than in worrying about things like dates on a calendar.  It was something else that made the whites feel superior.