Here is verse 12:
And you shall speak to him, saying, "So said the Lord of Hosts, saying: Behold a man whose name is the Shoot, who will spring up out of his place and build the Temple of the Lord.
and now 13:
And he shall build the Temple of the Lord, and he shall bear glory. And he shall sit and rule on his throne, and the priest shall be on his throne. And a counsel of peace shall be between them [both].
So the pronouns in verse 13 point to the person identified in verse 12. On verse 12, Rashi writes:
whose name is the Shoot: He is Zerubbabel, mentioned above (3:8): “Behold, I bring My servant, the Shoot,” since his greatness burgeoned little by little. Some interpret this as referring to the King Messiah, but the entire context deals with the [time of the] Second Temple.