According to Lutherans, what is the Kingdom of God?

What your American Lutheran is probably really driving at is that the US Constitution, especially in its First Amendment, insofar as it seeks to effect a separation between "Church" and "State" is in harmony with Scripture.

The First Amendment grants five Rights, which are Freedom of Religion, Freedom of Speech, Freedom of Press, Freedom of Assembly and Freedom of Petition of Grievances.

For more info on each one, click on each of these here:- https://en.wikipedia.org/wiki/List_of_amendments_to_the_United_States_Constitution

But this view he would hold, despite the fact that the church which Martin Luther helped to set up did not separate Church and State. In Luther's day it was the Anabaptists who believed in, and started, fellowships of ecclesia ("called out") voluntary believers in Christ... that is, "called out" from the surrounding unbelieving society. Luther and all the Reformers believed in State/Churches essentially they continued with the church structure just as they received it from Roman Catholicism of those days... everyone in the community is to be treated as Christian, except those who reject the church/state union who are to be exterminated as heretics.

For more on this see "The Reformers and their Stepchildren" by Leonard Verduin, published in 1964.

This is what the Lutheran Church—Missouri Synod, has to say about the Kingdom of God:

Kingdom of God: Term prominent in proclamation of John the Baptist, Jesus, and apostles; evidently synonymous with “kingdom of heaven.” Formerly it was taken for granted that “kingdom of God” is equivalent to “Christian ch.”; contemporary scholars hold that “kingdom” usually means dominion or rule, and that only in a derived way, by figure of speech, does it in some passages designate subjects in God's kingdom. When Jesus said that the kingdom of God had come near (Mk 1:15), he announced that God was, through the Messiah, laying the foundation for His gracious rule in human hearts. This rule presupposes that forgiveness of sins has been procured and that people accept it in faith. Where there is such acceptance, God has entered the heart and governs human thoughts and actions. Those who heard the message of John and of Christ were informed that God was preparing something special, that the fullness of the time had come (Gl 4:4), and that the plan of God for man's salvation was now to be carried out. The term did not point to an external kingdom like that of David or Solomon or the Roman Empire, but to something spiritual, the gentle rule of God through the Holy Spirit in the hearts of men. Passages which speak of the rule of God include Mt 12:28; Mk 4:11; Lk 9:27; 11:20. Passages which speak of the sum total of the subjects include Mt 13:41; 16:19. The kingdom is at times spoken of as a future blessing (Mt 7:21; 8:11), at times as a present reality (Lk 16:16; 17:20; Jn 3:3–5). It should be our heart's desire to be under the gracious rule of God (Mt 13:44–46).

This is what the Evangelical Lutheran Church in America has to say about politics, earthly princes, and ‘Faith in the Left-Hand Kingdom’:

[2] The familiar Lutheran distinction between the two kingdoms is useful here. God rules the right-hand spiritual kingdom through the gospel and grace and the left-hand secular kingdom through law and the sword. The spiritual kingdom is an internal kingdom existing within the hearts of Christians. It produces true righteousness. The earthly kingdom is an external kingdom that exists in the rest of the world. It does not produce righteousness but only peacefulness. Since the spiritual kingdom is a kingdom of faith, the earthy government, which effects only our external actions, has no authority there. And since the earthly kingdom is--according to Luther--largely a kingdom of unbelievers, the earthly prince should not try to rule it through the Gospel.

[3] Since rule in the earthly kingdom is not based on the Gospel, there is no reason why the earthly prince must necessarily be a Christian and consequently no reason why we, as voters, should make our support of candidates conditional upon their profession of Christian faith. We are not electing pastors or bishops; we are electing presidents and governors, and non-Christians can fill these offices as well as Christians.

[4] However, while Luther recognized that an earthly prince did not necessarily have to be a Christian, he also saw an advantage in the prince being a Christian. When the government wields the sword properly -which means solely for the purpose of restraining the wicked and protecting the good -it preserves peace on earth and is a great benefit to all, Christian and non-Christian alike. When the sword is used wrongly, to make the prince powerful or rich, everyone suffers. Unfortunately, the power for the sword proves a great temptation for most mortals. In On the Secular Authority, Luther wrote that most princes are of the latter kind, selfish and corrupt. Selfless princes are a "miracle… not impossible, but quite unusual." According to Luther, the only hope of having a selfless prince is when the prince is a Christian who has the Holy Spirit within. A Christian prince will approach the office as an opportunity to serve others rather than as an opportunity for self-aggrandizement. Such righteous behavior can be expected of "No one but a real Christian and one who is full of the Spirit…. It cannot be done without grace."

Because I am not a Lutheran and I know absolutely nothing about American politics, I would not presume to comment on what that gentleman meant when he said “the United States is not included among the kingdoms of the inhabited earth because the USA is of the Kingdom of God.” However, from what Luther had to say about God ruling the right hand (spiritual) kingdom through the gospel and grace and the left-hand secular kingdom through law and the sword, I would take that to imply that God allows secular governments and authorities to rule and they are “God’s servant, an agent of wrath to bring punishment on the wrongdoer” (Romans 13:4).

It seems to me that Lutherans perceive politicians and rulers as “princes” and that these “earthly princes” have no authority over the spiritual Kingdom of God.