Q. What was Baby Jesus ' redeemed with ' at the Temple?
[Luk 2:21-24 NASB95] [21] And when eight days had passed, before His circumcision, His name was [then] called Jesus, the name given by the angel before He was conceived in the womb. [22] And when the days for their purification according to the law of Moses were completed, they brought Him up to Jerusalem to present Him to the Lord [23] (as it is written in the Law of the Lord, "EVERY [firstborn] MALE THAT OPENS THE WOMB SHALL BE CALLED HOLY TO THE LORD"), [24] and to offer a sacrifice according to what was said in the Law of the Lord, "A PAIR OF TURTLEDOVES OR TWO YOUNG PIGEONS."
The questioner is confused because s/he is looking for direction in Exodus. The direction given in Exodus 13 only applied to the role of the first born males, who at that time served as a quasi-priesthood. It wasn't until the establishment of "the sanctuary" in Leviticus that the tribe of Levi assumed the priesthood established the relevant "redemption price" for firstborn human males, to "pay off" the LORD for not murdering the firstborn of the Jews along with the murdering of the Egyptians' first born sons:
[Exo 13:2 NASB95] [2] "Sanctify to Me every firstborn, the first offspring of every womb among the sons of Israel, both of man and beast; it belongs to Me."
So every first born Jewish parent became obligated to give their firstborn son for priestly service because the Death Angel (IE: the preexistent Messiah?) saw the blood of the innocent animals and "passed over" the first born of the Jews.
But the sacrifice of the Passover was insufficient to release the Jews of their debt to the LORD. The progeny of the Jews were forever obliged to the LORD to pay God back for his mercy and grace in not permitting the slaughtering of the Jews. By saving the parents, the LORD saved all future generations, so all Jews will forever be obligated to pay the LORD a "salvation fee" to the LORD in the form of either perpetual servitude as a priest or the blood of a flock animal.
[Num 18:14-16 NASB95] [14] "Every devoted thing in Israel shall be yours. [15] "Every first issue of the womb of all flesh, whether man or animal, which they offer to the LORD, shall be yours; nevertheless the firstborn of man you shall surely redeem, and the firstborn of unclean animals you shall redeem. [16] "As to their redemption price, from a month old you shall redeem them, by your valuation, five shekels in silver, according to the shekel of the sanctuary, which is twenty gerahs.
NONE of the above is relevant to Luke 2, invalidating the answers previously provided.
Related:
https://jewishencyclopedia.com/articles/6138-first-born-redemption-of#anchor5
What IS relevant is the cost to a woman for bearing a son. Childbirth involves the shedding of blood, and blood is taboo for ancient, and even modern, Jews. One has to absolve the mother of shed blood, even if it is her own:
[Lev 12:2-7 NASB95] [2] "Speak to the sons of Israel, saying: 'When a woman gives birth and bears a male [child,] then she shall be unclean for seven days, as in the days of her menstruation she shall be unclean. [3] 'On the eighth day the flesh of his foreskin shall be circumcised. [4] 'Then she shall remain in the blood of [her] purification for thirty-three days; she shall not touch any consecrated thing, nor enter the sanctuary until the days of her purification are completed. [5] 'But if she bears a female [child,] then she shall be unclean for two weeks, as in her menstruation; and she shall remain in the blood of [her] purification for sixty-six days. [6] 'When the days of her purification are completed, for a son or for a daughter, she shall bring to the priest at the doorway of the tent of meeting a one year old lamb for a burnt offering and a young pigeon or a turtledove for a sin offering. [7] 'Then he shall offer it before the LORD and make atonement for her, and she shall be cleansed from the flow of her blood. This is the law for her who bears [a child, whether] a male or a female.
But if one was poor, they could satisfy their blood guilt, and indebtedness for the survival of their generation, with something much less costly than a whole flock animal: a pair of birds:
[Lev 12:1-7 NASB95] [1] Then the LORD spoke to Moses, saying, [2] "Speak to the sons of Israel, saying: 'When a woman gives birth and bears a male [child,] then she shall be unclean for seven days, as in the days of her menstruation she shall be unclean. [3] 'On the eighth day the flesh of his foreskin shall be circumcised. [4] 'Then she shall remain in the blood of [her] purification for thirty-three days; she shall not touch any consecrated thing, nor enter the sanctuary until the days of her purification are completed. [5] 'But if she bears a female [child,] then she shall be unclean for two weeks, as in her menstruation; and she shall remain in the blood of [her] purification for sixty-six days. [6] 'When the days of her purification are completed, for a son or for a daughter, she shall bring to the priest at the doorway of the tent of meeting a one year old lamb for a burnt offering and a young pigeon or a turtledove for a sin offering. [7] 'Then he shall offer it before the LORD and make atonement for her, and she shall be cleansed from the flow of her blood. This is the law for her who bears [a child, whether] a male or a female.
So in Luke's narrative, Maria Carry was cleansed from her blood guilt by a token sacrifice (because she couldn't afford the actual cost of her indiscretion of spilling her blood while bearing the progeny of her forefathers, for being spared the death of their race):
[Lev 12:8 NASB95] [8] 'But if she cannot afford a lamb, then she shall take two turtledoves or two young pigeons, the one for a burnt offering and the other for a sin offering; and the priest shall make atonement for her, and she will be clean.'"
Luke is a much more sophisticated writer than is Matthew. In Matthew's portrait of Jesus as "King of the Jews," poverty was not an issue for the Boy Born King. All of the spiritual advisors from Iran did what they still do, and that is to generously fund the destroyer of the Jews:
[Mat 2:11 NASB95] [11] After coming into the house they saw the Child with Mary His mother; and they fell to the ground and worshiped Him. Then, opening their treasures, they presented to Him gifts of gold, frankincense, and myrrh.
Matthew's Jesus is not destitute. We see the differing priorities here also:
[Mat 5:3 NASB95] [3] "Blessed are the poor in spirit, for theirs is the kingdom of heaven.
Versus:
[Luk 6:20-21 NASB95] [20] And turning His gaze toward His disciples, He [began] to say, "Blessed [are] you [who are] poor, for yours is the kingdom of God. [21] "Blessed [are] you who hunger now, for you shall be satisfied. Blessed [are] you who weep now, for you shall laugh.
So unlike Matthew, who wants very badly to impress with depictions of Christ as royalty, Luke wants to portray Christ as "Joe the Plumber," poor as dirt and proud of it:
https://www.youtube.com/watch?v=nlcIKh6sBtc
Be mindful of which gospel you choose. Lukans tend to be much better people than Matthewans.