Some elements in Matthew and Luke’s infancy narratives are in agreement, which may indicate these elements are historical. Other elements though were chosen to communicate particular truths from Matthew and Luke to their intended audiences, which were different.

As Raymond Brown points out, both Matthew and Luke trace Jesus’ genealogy through Joseph, which would make Jesus a descendant (son) of David, an important title which the prophets of Israel predicted that the expected Messiah would have. Both authors also indicate that the Holy Spirit played a key role in Jesus’ conception; hence, Mary’s virginity. Both accounts include a proclamation by an angel. In both descriptions, the news of Jesus’ birth is shared with others. In both accounts, some kind of trouble either occurred or was predicted (e.g., the massacre of the innocent children, lack of room in the inn, prediction that Jesus would cause the rise and fall of many in Israel, etc.)

So much more to say but maybe this will inspire readers to compare the 2 accounts for themselves.