The most likely reason that a slaveholder would do this would be to keep the slave so ignorant as to be somewhat dehumanized. Slaveholders would not have wanted their slaves to have any sort of feeling that they were worth as much as white people were. If you deny a person the right to have even such basic things as a knowledge of their birth date, you are in some way telling them that they do not count nearly as much as a regular person does. Slaveholders would have wanted slaves to feel that way so that they would be more likely to accept their status in life.
It is worse than that. African-Americans were not allowed to practice their religion, speak in thier native tongue, or keep their names. Here is the evidence. When slavery ends, where do you go to work if you were a slave? What is your first and last name? Most often, free slaves would take the last name of the owner of the location where they worked; therfore, there are many African Americans named Johnson, or Washington, or Smith. Clearly, those are not African names, right? Malcolm X recognizes this. His real name was Malcolm Little, but he considered Little his family's slave name. He takes X instead. Why X? If you remember your math class X is the variable, the unknown.