I think all of us who have grown up in America has been robbed of something important. We never fully learned about kingdoms because we haven't lived in one, and while we have the basic concept we have not understood what it means to be a citizen of a kingdom.
When we say God is our King, we are acknowledging that He is the sovereign, the One who has ultimate say over us. Our lives are marked by our willful submission and obedience to His laws. On top of comparison to an earthly kingdom, in God's world, we were made by Him, we belong to Him and we are indebted to Him. Obviously many deny that and neglect Him in all these ways, but that doesn't change the facts. In a kingdom, such as England for instance, you can deny the sovereign of the Queen and refuse to submit, but that doesn't make her any less the Queen.
Something else totally unique about God's kingdom, not only do we live in the kingdom, but "the kingdom of God [lives] within [us]" (Luke 17:21). If we only acknowledge the first part, that we live in His kingdom, we misunderstand the kingdom as God intends it to be. God's saints make up His kingdom, and it is only alive and well because it lives in us. It's not something to see and touch or a place we travel to, and unless we understand the nature, we don't understand the kingdom.
Jesus's physical followers on earth misunderstood initially too, they were looking for a physical kingdom, with power, a kingdom to topple the earthly kingdoms which they lived in. Likely they would envision an army, and rule by conquering, the type of kingdom where a sword is yielded and the citizens subject themselves to the sovereign. Jesus tried to explain to them that His kingdom was different, that they could not physically pinpoint it or see it and it didn't have geographical boundaries (Luke 17:20-21). He told them that they were not physically born into it, and explained that to enter they must be "born again" (John 3:5-7).
What they understood in the physical, Jesus had to teach them to understand in the spiritual, but they at least understood the concept of a kingdom, and I think we in America have missed out not having lived in an earthly kingdom under an earthly ruler. We have always lived in a democratic type of society, (a constitutional republic to be exact) where the people have the power. Maybe I'm wrong, but I think that mentality makes it just a little tougher to understand yielding our will to that of the King.
Today, if you are a Christian, you are in His kingdom, and His kingdom is within you. It is a kingdom right now, established on the day of Pentecost as described in Acts 2, and it is also a kingdom yet to be fully realized pending the day of judgment and entrance into heaven. It has terms of entrance, and its citizens are a select group, made legitimate by God Himself through Jesus His sovereign son.
I do believe that if we had been born into an earthly kingdom we could better understand our place in the spiritual kingdom, but certainly, we are given sufficient information in scripture to understand what God's purpose for mankind is. God has established a kingdom whose inhabitants submit ourselves to our creator and recognize not only the rule and authority of God in our lives but our place in relation to Him.