The kingdom of Israel was established by God with Israel at Sinai with God as their king. It was over 400 years later that earthly kings ruled as deputies and representatives of the true king and God of Israel - the LORD Himself.
While David and his successors were earthly kings, they were to recognize that the real king of Israel was God. 1 Sam 8:7, 8, 24:6, 2 Sam 19:21, 1 Chron 28:5, 2 Chron 9:8, 13:8, Ps 5:2, 44:4. See also 1 Sam 12:14.
It is a simple matter of history that David’s descendants were not always faithful and the earthly Davidic dynasty ended in 586 BC with the final capture of Jerusalem by Nebuchadnezzar. However, the New Testament calls Jesus Christ, Messiah, “the Son of David” as a direct fulfilment of the (ultimately) eternal throne of David which Jesus inherited. Matt 1:1, 20, 9:27, 12:23, 15:22, 20:30, 15, 21:9, 15, Mark 10:35, Luke 1:32, 33, 18:38, 39, John 1:49, Acts 13:32-37, Heb 1:8. See also Rev 11:15, 19:16. Such a Messiah was prophesied long ago: Ex 15:18, Ps 10:16, 61:7, 68:16, 92:8, 93:5, 146:10, Isa 9:7, 47:7, Lam 5:19, Micah 4:7, etc. Compare Isa 55:3 with Acts 13:34 and John 1:49.
Note especially, what the angel said to Mary before Jesus’ birth in Luke 1:32, 33 –
He will be great and will be called the Son of the Most High. The Lord
God will give Him the throne of His father David, and He will reign
over the house of Jacob forever. His kingdom will never end!”
We see many times in the NT that Jesus was appointed the king of the Christian community with His enthronement symbolically portrayed in Rev 5, following His resurrection. Thus, Jesus said to sit on the throne of heaven at the right hand of God, Matt 26:64, Mark 14:62, 16:19, Luke 22:69, Acts 2:33, 7:55-56 (standing), Rom 8:34, Eph 1:20, Col 3:1, Heb 1:3, 8:1, 10:12, 12:2, 1 Peter 3:22. See also Ps 110:1, Matt 22:44, Mark 12:36, Acts 2:34, Heb 1:13, Rev 5.
Now, here is the "difficulty": The fact that Jesus is king does not mean that He is yet king of all the earth as many people do not acknowledge Him as king (that is obvious). However, that will occur one day when Jesus returns according to Rev 11:15 -
The seventh angel sounded his trumpet, and there were loud voices in
heaven, which said: “The kingdom of the world has become the kingdom
of our Lord and of his Messiah, and he will reign for ever and ever.”
This appears to be the predicted fulfillment Ps 110:1. of The NT uses exactly the same language as Ex 19:6 to describe the Christian kingdom:
“But you are a chosen people, a royal priesthood, a holy nation, a
people for God’s own possession, to proclaim the virtues of Him who
called you out of darkness into His marvelous light. Once you were not
a people, but now you are the people of God; once you had not received
mercy, but now you have received mercy.” 1 Peter 2:9, 10 (Compare Ex
19:5, 6, Hos 1:9, 4:6).
Let there be no doubt that the NT regards Jesus, Lord Messiah of the New Covenant as King and High Priest of the Christian community as the fulfillment (or more correctly the Messianic development) of the OT prophecies about God's kingdom.