This is a description of the fulfillment of kingly mediation that comes in the OT prophecies of the Davidic Messiah.
Kingly mediation would find fulfillment in that God promised David that he would place a son of his on his throne "endure forever before me" and "your throne will be established forever" (2 Sam. 7:16). This was a fulfillment of Israel's prophecy to Judah that eternal kingship would come from his line (Gen. 49:10). Much like Josiah death as a representative of the sins of Israel, Judah offers himself as a substitute for his brother Benjamin earlier in the narrative of Genesis (Gen. 44:33). Just as Isaiah prophesies about the fulfillment of the prophet like Moses, he also emphasizes the fulfillment of the Davidic Messiah. The Davidic Messiah, much like the figures prophesied to fulfill priestly and prophetic mediation, takes on divine qualities. He is described as "Immanuel" (7:14) that is, "God with us." In chapter 9, he is also described as "great light." This is more than reminiscent of the Servant of YHWH in chapter 49, being described as a "light to the nations," that we earlier connected with the divine kavod. Later, in the same chapter he is described as "Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace" (v. 6). The adjectives "Mighty" (בֹּ֔ורגִ) and "Everlasting" (same word with "Father" Hebrew, אֲבִיעַד) are only predicated of YHWH elsewhere in the Old Testament.
There also might be connection between the Messianic figure of chapter 9 and the Angel of YHWH. The LXX translates the verse not as "Wonderful Counselor" but rather as the "Angel of Great Council." There is little in the Hebrew text that would definitely suggest this translation, nevertheless, it is highly suggestive that Second Temple Jews connected these texts with the other texts that we have previously discussed that connect the Messiah with the Angel of YHWH.
This divine identity of the Messiah is suggested elsewhere in the prophets. Ezekiel describes the situation thus: "'My servant David will be king over them, and they will all have one shepherd" (Ezek. 37:24, Emphasis added). Earlier, God states that he will shepherd Israel: "I will shepherd the flock with justice" (34:16). Though text does not appear to explicitly teach a divine Messiah, what seems to be implicit is that because there is only one shepherd, God and the Davidic Messiah, are one person. It should also be noted that Ezekiel's prophecies of the Messiah are connected with the coming of a new covenant ("I will make a covenant of peace with them" 34:25, an "everlasting covenant" 37:26) and the divine act of cleansing from sin ("I [will] cleanse you from all your sins" 36:33), which connect it to the prophecies of Isaiah 53, 61, Jeremiah 31 and Daniel 7, 9.
Moving on to Isaiah chapter 11, the Davidic Messiah is described as "A shoot will come up from the stump of Jesse; from his roots a Branch will bear fruit" (11:1). This is very similar language to what we find elsewhere in the Old Testament. We read in Jeremiah 23 that the David Messiah who is also referred to as a "Branch" (v. 5). His name will be "The LORD Our Righteousness" which again, suggests a divinity. The language of "Branch" and "Shoot" is remarkably similar to that used in Isaiah 53: "He grew up before him like a tender shoot, and like a root out of dry ground" (Isa 53:2). The common wording of these passages suggests then that the "Branch" is the same person as the Servant. On a typological level as well, it also makes sense that the Davidic Messiah would be connected with the Servant, who acts as a new Passover lamb for a new exodus. Previously, in the case of Judah and Josiah, the Davidic line has acted as a substitute for others.
The description of the "Shoot" coming "out of dry ground" also appears to connect Isaiah 53 both with Isaiah 7 and Genesis 2. We are told that Adam was taken from the ground before it had rained on the earth (Genesis 2:6). This makes the Davidic Messiah a new Adam in that he takes over Adam's position in Genesis 1:28. Similarly, we are told in Isaiah 7 that "Immanuel" will be born of a "virgin." Ground that has not been watered might very well be a metaphorical way of talking about virginity. This not only connects the Messianic prophecies of Isaiah 7, 9, 11 to the Servant songs, but also connects them to the protevangelium ("seed of the woman" being strongly suggestive of virgin birth, as we noted earlier), but also the prophecies of Daniel 7, 9, 10, which we have previously suggested have a direct connection to the Servant songs.