The historical setting for Isaiah 7 concerns the efforts of Syria (Aram) and Israel (Ephraim, the northern kingdom) to form an alliance to attack king Ahaz of Judah (cf. 2 Kings 15:37). Some refer to this as the Syro-Ephraimite coalition against Judah.
The southern kingdom over which Ahaz ruled is referred to as the “house of David” twice in this chapter (7:2,13), and again in Isaiah 22:22.
In the days of Ahaz the son of Jotham, son of Uzziah, king of Judah, Rezin the king of Syria and Pekah the son of Remaliah the king of Israel came up to Jerusalem to wage war against it, but could not yet mount an attack against it. When the house of David was told, “Syria is in league with Ephraim,” the heart of Ahaz and the heart of his people shook as the trees of the forest shake before the wind. (Isaiah 7:1-2 ESV)
This chapter clearly mentions three distinct nations or governments – Syria, Israel, and Judah (the house of David). According to the Bible, the throne of David was established forever (2 Samuel 7:11-15). Christians understand that Jesus Christ is the final fulfillment of that promise (Luke 1:31-32; Acts 2:25-36).
Three fragments of an Aramic inscription were found at Tel Dan in 1993/1994 mentioning the king of Israel, the “house of David” and the Aramean (Syrian) god of storms and warfare, Hadad. You may recall that some of the Syrian kings were named after Hadad (e.g., Ben-Hadad, 1 Kings 15:18). This document probably refers to a different historical setting, but the kingdoms are the same: Israel, Syria, and Judah (house of David). Because all of the kings of Judah from David to the carrying away into Babylon are of the family of David, It is appropriate to refer to the house of David as the Davidic dynasty.
The Tel Dan inscription is displayed in the Israel Museum in Jerusalem.