Only the very poor were permitted to offer two doves or pigeons when presenting a newly born son at the temple, as Joseph and Mary do in Luke 2:22. On the other hand, the gifts of gold, frankincense and myrrh would have made them quite rich. There are two possible reasons that Joseph and Mary only offered pigeons or doves. Either the magi had not yet brought their valuable gifts to Bethlehem, or the two nativity stories are independent accounts in which at least some details are not entirely historical.
A problem with the first option is that we are told the young family left Jerusalem after the presentation at the temple, returning to Nazareth and apparently never returning to Bethlehem, which indeed they had no reason to do. Implicitly, they remained in Nazareth, because we are told they travelled to Jerusalem each year for the Passover:
Luke 2:39-41: And when they had performed all things according to the law of the Lord, they returned into Galilee, to their own city Nazareth. And the child grew, and waxed strong in spirit, filled with wisdom: and the grace of God was upon him. Now his parents went to Jerusalem every year at the feast of the passover.
Raymond E. Brown, in An Introduction to the New Testament, page 114, provides support for the second option, because he says Luke's infancy narrative is massively different from Matthew's and in details is virtually irreconcilable with it. There is no suggestion that the young couple had ever lived in Nazareth; in fact, quite the opposite. On returning from presumably a few years in Egypt, Joseph and Mary head back to their house in Bethlehem but, being warned by an angel, turn aside and travel to Galilee, which is beyond the reach of Archelaus. In Galilee, they went and dwelt in a city called Nazareth, a decision that Matthew says explains why Jesus is called a Nazarene:
Matthew 2:22-24: But when he heard that Archelaus did reign in Judaea in the room of his father Herod, he was afraid to go thither: notwithstanding, being warned of God in a dream, he turned aside into the parts of Galilee: And he came and dwelt in a city called Nazareth: that it might be fulfilled which was spoken by the prophets, He shall be called a Nazarene.
In Matthew's account, the flight to Egypt was sudden and unexpected, so Joseph and Mary had no chance to present Jesus at the temple and make the offerings appropriate to their new economic status, and in any case they now knew that Jerusalem would be too dangerous a place to take Jesus.