The first record of this effort at harmonization appears to come from Annius of Viterbo (1432-1502). His guess is plausible, yet not a demonstrable fact.

The basis for his notion is the difference in perspective in the infancy narratives. Matthew reflects on Joseph’s experience as a man who gets guidance from God in his dreams; while Luke reflects on Mary’s experience as a young woman in conversation with an angel. This interest in Mary might mean that Luke would decide to trace Mary’s ancestry, instead of Joseph’s.

Yet is that what he wanted to do? It is not obvious. Luke did make the point that Mary was Jesus’ only human parent (1:34-35); he also noted that Jesus was the son “as was supposed” of Joseph (3:23). Since Luke had made both those statements, there is no reason for supposing he could not have said, “Mary, daughter of Heli, son of Matthat” if that’s what he intended.

But instead, he said he was tracing Joseph’s ancestry. Throughout the genealogy Luke’s wording is very direct: “Jacob, son of Isaac, son of Abraham” (3:34) is clearly a line of sons begotten by their fathers. And “Joseph, son of Heli, son of Matthat” (3:23-24) looks like the same thing: a line of sons begotten by their fathers, rather than a hidden way of saying “Mary, daughter of Heli.”

As you note, the two genealogies do not say the same thing. We might simply concede this difference, and acknowledge we don’t know why the two don’t match. But people sometimes try to protect the accuracy of the Bible by asserting that one of the genealogies reflects Mary’s ancestry rather than Joseph’s. Yet how does this help? If Luke has actually given us Mary’s genealogy while declaring it the lineage of Joseph, how does that save us from an inaccurate text?