In order for the child Prince to act as another Intercessor, or Advocate just like Christ, the child Prince must also function as High Priest as well:

Zechariah 6: 9-13 The word of the Lord came to me: “Take silver and gold from the exiles Heldai, Tobijah and Jedaiah, who have arrived from Babylon. Go the same day to the house of Josiah son of Zephaniah. Take the silver and gold and make a crown, and set it on the head of the high priest, Joshua son of Jozadak. Tell him this is what the Lord Almighty says: ‘Here is the man whose name is the Branch, and he will branch out from his place and build the temple of the Lord. It is he who will build the temple of the Lord, and he will be clothed with majesty and will sit and rule on his throne. And he will be a priest on his throne. And there will be harmony between the two.’

The book of Zechariah makes it clear that the high priest Joshua is only symbolic of this future Prince and High Priest whom God also calls the Branch:

Zechariah 3: 8-9 “‘Listen, High Priest Joshua, you and your associates seated before you, who are men symbolic of things to come: I am going to bring my servant, the Branch. See, the stone I have set in front of Joshua! There are seven eyes on that one stone, and I will engrave an inscription on it,’ says the Lord Almighty, ‘and I will remove the sin of this land in a single day.

Now many will object and say that Christ is this prophesied Branch, and not the Holy Spirit! However, if you examine closely all the prophecies regarding the Branch, you will see that 2 Branches and not just 1 are always referred to:

Isaiah 11: 1-2 And there shall come forth a Rod out of the stem of Jesse, and a Branch shall grow out of his roots: And the spirit of the Lord shall rest upon him, the spirit of wisdom and understanding, the spirit of counsel and might, the spirit of knowledge and of the fear of the Lord;

Note that the Rod coming forth out of the stem (or trunk) of Jesse is different from the Branch that grows out of his roots, since the Rod and the Branch appeared on two different parts of the tree, namely the stem and the roots. The stem is by definition the main body or stalk of a plant or shrub, typically rising above the ground. The root on the other hand is by definition the part of a plant that attaches it to the ground or to a support typically underground. Therefore this prophecy by Isaiah applies equally to both the Rod and the Branch, which symbolize Christ and the Holy Spirit respectively.