It is a mitzvah (to sanctify the first born to G-d) that every first born male child is ritually given to G-d. It is NOT a prophecy (let alone a messianic prophecy). Every first born Jewish male child is redeemed. Neither Sh'mot / Exodus 13:2 or Luke 2:23 are prophecies.
Luke 2:23-25 says “as it is written in the Law of the L-rd, “When the time came for the purification rites required by the Law of Moses, Joseph and Mary took him to Jerusalem to present him to the L-rd as it is written in the Law of the L-rd." Every firstborn male is to be consecrated to the L-rd”, and to offer a sacrifice in keeping with what is said in the Law of the L-rd: “a pair of doves or two young pigeons.”
Sound like prophetic fulfillment to you?
“Every first issue of the womb of any creature, which they present to the L-rd, whether of man or beast, shall be yours. However, you shall redeem the firstborn of man, and the firstborn of unclean animals you shall redeem.” Bamidbar / Numbers 18:15.
This is פדיון הבן / pidyon haben. A Jewish firstborn son is "redeemed" by use of five silver coins from his birth-state of sanctity. The anonymous author of Luke says nothing about the five shekels.. See Numbers 3:47, 18:15; Leviticus 27:6 – it is 5 silver shekels for the priest, not a sacrifice of two birds.
A mother, post child-birth, is ritually impure for a set period of time. When there is a Temple standing and the woman’s “purification period for a son or a daughter is complete, she shall bring to the priest, to the Communion Tent entrance, a yearling sheep for an עֹלָ֖ה / olah, and a young common dove (pigeon), or a turtle dove for a חַטָּאת / cḥattat (accidental sins).” Vayikra / Leviticus 12:6. Notice that Luke gets it wrong. If the parents cannot afford a sheep, a pigeon can be substituted for the burnt offering and this appears to be what Jesus’ parents did – they brought two birds. Luke does not say that the sacrifices are for Mary’s ritual purity – most Christians probably think the sacrifices were for the baby, when in reality they have to do with the ritual purity of the mother “in keeping with what is said in the Law of the L-rd (they sacrificed): “a pair of doves or two young pigeons.” This brings a serious problem for Catholics who claim that Mary was an immaculate conception (meaning she, herself, was "without sin"). They believe Mary was "free from original sin." If that is the case, why is she bringing a sin sacrifice for herself?
A woman, having given birth, brought two sacrifices for herself:
There is a much bigger problem with Luke 2:22-24. It is totally contradicted by Matthew 2:19-23. Matthew says that Jesus was born in Bethlehem and then an angel told Joseph “take the child and his mother and escape to Egypt. Stay there until I tell you, for Herod is going to search for the child to kill him. So he got up, took the child and his mother during the night and left for Egypt” Matthew 2:13-14.
From Egypt Joseph went to Nazareth “he went and lived in a town called Nazareth.” Matthew 2:23. They don’t go to Jerusalem at all! There is no tale of the redemption of the firstborn or the purification of Mary. . .
Matthew and Luke are in total contradiction to each other. Matthew clearly says that Joseph did not return to Judah from Egypt in Matthew 2:19-23.
The biggest problem is that Luke 2:22-24 is not a messianic prophecy.
One would expect that if Jesus was the first son of Mary and Joseph that the parents would pay the 5 shekels for the redemption of the firstborn son and bring the necessary sacrifices for Mary’s ritual purity. Mary’s purity sacrifices and Jesus’ redemption as the first born son is done for every firstborn Jewish boy. This practice continues to this very day. Link.
Luke 2:22-24 is not a messianic prophecy fulfillment. Sh’mot / Exodus 13:2 is simply speaking of the redemption of the first born Jewish son of a Jewish mother – all firstborn sons, not specifically the messiah – it is not a messianic prophecy, and neither is Luke 2:22-24.