It is used as an artifact to show Lelouch's interest in freedom and democracy and all that stuff we usually associate with the "United States" image. And I am not talking about USA. In our world, there are/were a lot of countries that used the term "United States" in their name.
In Code Geass universe, there are no USA (now I am talking about THAT stars and stripes real world nation).
the duke of Britannia bribed Benjamin Franklin to betray Washington, and the rebellion failed.
Then Britannia loses the british isles, and relocates to the New World.
in 1807 AD, Elisabeth III is captured by a revolutionary militia and is forced to abdicate. She and her knights move Britannia into the new world.
But Japan has massive sources of Sakuradite, a scarce resource in the west.
Sakuradite is known since the middle ages, but its too scarce in the west. Marco Polo in his travels learns that Japan has huge deposits, and this knowledge spreads in the west.
So in order to fuel the war machine, it is vital for Britannia to hold unto Japan.
Since Britannia holds (at the time of R2) almost all of the world and it is a monarchic Empire, Lelouch wants to declare at the same time his rejection of the government system of his father, and his intent with building a free nation.