The meaning of "Tell me about yourself" is "describe yourself". The person does not respond by "telling himself or herself" a person is not a thing that can be told. Instead the response is to tell some items of description, that is some items about the person.
Similarly one does not say "Tell me New York". Rather, one says "Tell me about New York." One uses the form "Tell me X" only when X is i thing that can be told, such as "a story". Otherwise, one uses "Tell me about X" no matter what X is. That is just how English handles this -- "Tell me about X." means "describe X.", while "Tell me X" is meaningless unless X is an account in words:
- Tell me what happened.
- Tell me the plot.
- Tell me about the movie.
- Tell me what she said.