Yes that is what they teach american kids: you pay the fee, you own the stuff.
It often turns out to be more complicated, because most of the land in the United States is stolen land. In the Hawaiian Kingdom, King Kamehameha III "invented" private property because the kingdom was starting to open up to the global world and he wanted to make sure the lands which the people occupied could not be stolen by way of foreign military invasion. Which is impressive considering that a foreign military invasion did indeed come, and those lands could not legally be seized by the invading forces. The land would need to be stolen by other means. Up until then, land was not property it was a responsibility, to the family, the community, and by extension, the nation as a whole. Land could not be owned, only stewarded. Land could not be sold for profit. A friend of mine put it nicely, he said that, to him, the buying and selling of land is like prostituting a family member, who is also an elder.
It just doesnt make sense, so try not to think about it. You get too many people thinking about it at the same time, the whole system is liable to fall apart.