“And when the days of her purifying are fulfilled, for a son, or for a daughter, she shall bring a lamb a year old for a whole burnt offering, and a young pigeon, or a turtledove, for a purification for sin offering, to the door of the tent of meeting, to the priest, and he shall offer it before Yahweh, and make atonement for her; and she shall be cleansed from her blood-flow. This is the law for her who bears a child, whether a male or a female.”
Once the woman had safely reached the end of her period of purification she was then to bring a one year old lamb (as for the daily sacrifice) and a young pigeon or turtledove to the priest for him to offer on the altar before Yahweh, ‘to make atonement for her'. This makes clear the connection with required atonement. And note the emphasis on her blood-flow. It is that primarily that has to be cleansed. By giving birth she has released blood, and that has made her unclean. But what it signified was also in mind.
The lamb was for a whole burnt offering. It was an act of gratitude, tribute, dedication and atonement. The bird was for a purification of sin offering. She needed forgiveness and reconciliation with God. By bringing her child into the world she had introduced further sin into the world and increased the burden of sin. She shared the responsibility of Eve.