At the beginning of the story, Desiree is no one. She is adopted by the Valmondes, and becomes one of them. Then she marries Armand Aubigny, and becomes his wife. She then bears a child and becomes a mother. The reader knows very little to nothing about Desiree herself other than how she fits amongst these characters. Chopin could have been portraying women of the time and lack of induviduality.
The baby is also never named other than "Desiree's baby." The baby has no identity other than to it's mother and father, the fact that it was a son, and the fact that somewhere in the baby's lineage there is African blood on the father's side.