I posted the question and in it I mistakenly associated the Lukan passage with the redemption of the firstborn male found in Exodus. The actual background of Mary's offering is found in Leviticus, most specifically verses 2 and 6-8:
[Lev 12:1-8 NLT] (1) The LORD said to Moses, (2) "Give the following instructions to the people of Israel. If a woman becomes pregnant and gives birth to a son, she will be ceremonially unclean for seven days, just as she is unclean during her menstrual period. (3) On the eighth day the boy's foreskin must be circumcised. (4) After waiting thirty-three days, she will be purified from the bleeding of childbirth. During this time of purification, she must not touch anything that is set apart as holy. And she must not enter the sanctuary until her time of purification is over. (5) If a woman gives birth to a daughter, she will be ceremonially unclean for two weeks, just as she is unclean during her menstrual period. After waiting sixty-six days, she will be purified from the bleeding of childbirth. (6) "When the time of purification is completed for either a son or a daughter, the woman must bring a one-year-old lamb for a burnt offering and a young pigeon or turtledove for a purification offering. She must bring her offerings to the priest at the entrance of the Tabernacle. (7) The priest will then present them to the LORD to purify her. Then she will be ceremonially clean again after her bleeding at childbirth. These are the instructions for a woman after the birth of a son or a daughter. (8) "If a woman cannot afford to bring a lamb, she must bring two turtledoves or two young pigeons. One will be for the burnt offering and the other for the purification offering. The priest will sacrifice them to purify her, and she will be ceremonially clean."
In other words, this is about the cleansing of the mother from the ritual impurity due to contact with blood.
What she is commanded to bring to be killed by the priest and offered to the LORD is a yearling lamb and a single pigeon/turtledove. The offerings express her appeal for cleansing and a pleasing sacrifice of obedience to God. As prophecy, it indicates that the boy she has birthed, in the prime of his life will become a well pleasing sacrifice for sin, thereby providing cleansing for his people:
[2Co 5:21 NET] (21) God made the one who did not know sin to be sin [or, "a sin offering"] for us, so that in him we would become the righteousness of God.
[Eph 5:2 NKJV] (2) And walk in love, as Christ also has loved us and given Himself for us, an offering and a sacrifice to God for a sweet-smelling aroma.
[Phl 4:18 NKJV] (18) Indeed I have all and abound. I am full, having received from Epaphroditus the things sent from you, a sweet-smelling aroma, an acceptable sacrifice, well pleasing to God.
It also illustrates that with God, nothing shall be impossible, because God brought a clean thing from an unclean woman and in so doing fulfills the prophetic import of Job 14:
[Job 14:1-9 NET] (1) "Man, born of woman, lives but a few days, and they are full of trouble. (2) He grows up like a flower and then withers away; he flees like a shadow, and does not remain. (3) Do you fix your eye on such a one? And do you bring me before you for judgment? (4) Who can make a clean thing come from an unclean? No one! (5) Since man's days are determined, the number of his months is under your control; you have set his limit and he cannot pass it. (6) Look away from him and let him desist, until he fulfills his time like a hired man. (7) "But there is hope for a tree: If it is cut down, it will sprout again, and its new shoots will not fail. (8) Although its roots may grow old in the ground and its stump begins to die in the soil, (9) at the scent of water it will flourish and put forth shoots like a new plant.
[Job 14:1 KJV] (1) Man that is born of a woman is of few days, and full of trouble.
[Job 15:14 KJV] (14) What is man, that he should be clean? and he which is born of a woman, that he should be righteous?
[Job 25:4 KJV] (4) How then can man be justified with God? or how can he be clean that is born of a woman?
I'm not sure of the relationship between Paul's words in Galatians and those of Job 14:
[Gal 4:4 KJV] (4) But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law,
But all of this dispenses with the need and repudiates the invention of the Catholic dogma of "The Immaculate Conception" and instead fulfills the Psalm of David, which in turn casts doubt on the popular use of this as evidence of the Catholic dogma of Original Sin:
[Psa 51:5 KJV] (5) Behold, I was shapen in iniquity; and in sin did my mother conceive me.
However, Mary did not offer a yearling lamb but rather the alternative offering provided by the poor. This indicates that Joseph and Mary were poor and either they had not received the gifts of the Magicians yet or they did not consider it appropriate to use the gifts to provide her offering. So Jesus was born very rich:
[2Co 8:9 KJV] (9) For ye know the grace of our Lord Jesus Christ, that, though he was rich, yet for your sakes he became poor, that ye through his poverty might be rich. In his adulthood Jesus gave all his wealth to the poor.