In Matthew’s genealogy, the term “king” is used only once–for David in 1:6 (“and Jesse the father of David the king”). This singular occurrence may be important because the names after David and until the exile are all kings (Solomon, Rehoboam, Abijah, Asaph, Jehoshaphat, Joram, Uzziah, Jotham, Ahaz, Hezekiah, Manasseh, Amos, Josiah, Jechoniah), but the title “the king” is not repeated for any of these fourteen people.
The use of “the king” highlights David’s name in the genealogy. Matthew 1:1 opened with saying Jesus is “the son of David.” It’s important for Matthew’s purposes, then, to connect Jesus the Christ to David the King. The emphasis is not on the fact that Jesus is the son of Solomon the king or Rehoboam the king or Uzziah the king. The point is that Jesus is the “son of David” who is “David the king.” God had made covenant promises to King David (cf. 2 Sam 7), and in Jesus we see covenant fulfillment.
The connection between David and Jesus is made even clearer when, in Matthew 1:16, Jesus is called “Christ.” Jesus is God’s consummate Anointed One, Israel’s (and the world’s) Messiah, the King whose rule would never end.