This verse portrays a unique point in time in the end-times where barriers in the norm will have been crossed, including in the nature of sons of men so as to come a form.
The house of David is consistently a reference to David's dynasty relative other sovereigns that lead the two nations of Jacob's descendants.
2 Sa 3:1
Now there was long war between the house of Saul and the house of David: but David waxed stronger and stronger, and the house of Saul waxed weaker and weaker.
Having said that, the same David is a type for Messiah in scriptures as you well know, and for buttressing which concept there are two complementary scriptures that address the controversial aspect of Messiah being given the same attention as that to the Most High God in the term 'seek'.
Hosea 3:5 Afterward the sons of Israel will return and seek the LORD their God and David their king; and they will come trembling to the LORD and to His goodness in the last days.
As well as;
And ye shall seek me, and find me, when ye shall search for me with all your heart.
Seeking is rarely a term employed on a normal king, but to entities that are objects of worship, false gods. So the Most High directly attributes the same seeking from His people meant for Him as being concurrent with seeking of this king David, which David is said to come in the ''last days.'' A Messianic reference.
Since ‘houses’ were characterised by sons and servants, and the ancient David had quite a house, sons who were princes in Israel and heirs to his throne, so does this King David have sons in His followers, some of whom constitute the said house of David in Zechariah 12, while the rest of servants of God are signified simply as ‘inhabitants of Jerusalem.’
Still, these sons and servants of this ‘David’ being the work of God, therefore the house is also God’s house.
Thus says the LORD of hosts, 'If you will walk in My ways and if you will perform My service, then you will also govern My house and also have charge of My courts, and I will grant you free access among these who are standing here.
Again this being a Messianic reference, it's still that Messianic figure as God hasn’t houses of bricks and mortar, but dwells in contrite hearts below and holy beings above(Isaiah 57:15)
These are the humble beginnings of this ‘David.’
The house of God in the old setting was the temple of the Most High. In the new dispensation the House of God are the saints in whom is His spirit, a state going by the allegory of ‘called by His name.’
The exercise of choosing these 'sons' from among men by the Messianic figure in Psalm 45:16 is the building of the ‘temple of the LORD,’ in Zechariah 6 by ‘Joshua’ chosen to judge God’s House in Zechariah 3:7.
Notice that there are several ‘houses’ mentioned in Zechariah 12 but ONLY the house of David is said to be like the ‘angel of the LORD,’ it’s because they are the ruling sons of God from among men. There is a lot to say here but we touch on only a few aspects.
The term ‘son,’ and that of God, is thrown around alot, and in which case it’s original intent is missed. Sons of God are ONLY rulers, spirit princes in several capacities in all creation, elsewhere rendered as mighty ones, the mysterious 'elohim' appearing in scriptures where they shouldn’t, one such case being Psalm 8:5, causing confusion even among transcribers because of unfamiliar territory in the phrase ‘you made him lower than elohim.’
These are the type of sons in Job 1, 2 and Job 38, but most of whom went wayward in their ways, some of which ways are only alluded to in Genesis 6.
It’s to the stature of these mighty sons that the Most High ordains a Messianic figure in Psalm 45 to appoint from among men such ‘princes’ in all the earth, who would rule over entities over whom the devil had been prince prior to the point of Ezekiel 31:12 when his influence, alongside that of all other fallen sons or spirit princes, will be withdrawn from off all creatures under the heavens.
This is the nature of times and things that characterise this peculiar house of David.
The psalmist indicates the real meaning of ‘sons of God’ in Psalm 82, and qualifies the nature of these sons in mentioning how one of them fell, the ‘son of the morning’ in Isaiah 14.
6 I said, "You are gods, And all of you are sons of the Most High.
7 Nevertheless you will die like men And fall like any one of the princes."
These appointed by the Messianic figure in Psalm 45:16 are the house of David.
So wherever there’s mention of ‘sons of God,’ the name of the LORD is also an aspect, and which name is what characterises entities written about as ‘the angel of the LORD’
This is the same name that these sons of God are ‘called by'.
I will say to the north, Give up; and to the south, Keep not back: bring my sons from far, and my daughters from the ends of the earth;
Even every one that is called by my name: for I have created him for my glory, I have formed him; yea, I have made him.
Verse 7 is often read in a stampede, yet it says things that should only provoke thought, because creating and forming are unique references. This is because these sons eventually become marvels of creatures of glory, and which glory is that mentioned in Zechariah 12, having been men and now spirit princes, like the angel of the LORD.
In that day the LORD will defend the inhabitants of Jerusalem, and the one who is feeble among them in that day will be like David, and the house of David will be like God, like the angel of the LORD before them.
This ultimate form of these sons who are the house is attained progressively, that’s why we have creating, forming and making mentioned above, in that those that make up the house of ‘David’ start out as David also starts out, when He embarks on the epic journey of taking sons of God out men, this is the allusion to the ‘feeble’ one being like David.
Come the time when all the fallen princes have been kicked out of their principalities in all the earth, and these sons of God attained to glory and powers for which the Most High says He created and formed and made them, is the time when this house of David will be like ‘elohim,’ the angel of the LORD before them, that is, Messiah in another form.
This is the nature of things that Zechariah 12 foretells..