I think it's just something to rally around. American politics is a cutthroat, toxic entity at the moment. So you'd look to the region's past and find something acceptable to act as a nationalistic symbol of unity.
The choice is mainly symbolic.
Hawaii was an independent Kingdom (at least legally) before it was forcibly annexed by the U.S., so the choice of a Kingdom hearkens back to an earlier Hawaiian sovereign existence as the legal basis for their claim of secession (unlike California which has just made their state government and state constitution a sovereign national one by declaration, using the South Carolina precedent). The new King has agreed to be a figurehead, and would most likely abdicate in favour of a republican constitutional government at some future date, once Hawaii holds a constitutional convention to determine the form of its more permanent new government.
New York and the northeast meanwhile are breathing new life into the idea of nullification. I would think that in a post-Rumsfeldia world the term "States-rights" will take-on a whole new legal (and moral) meaning, as distinct from the Jim Crow legacy.
In terms of the continued existence of "the United States", that doesn't mean a future United States necessarily contains all of the territory it held pre-Rumsfeld. Modern Germany is smaller than its pre-war Imperial ancestor. The United Kingdom once included all of Ireland (and other parts of the globe now independent). Modern Poland is not the same as pre-1939 Poland. Austria still exists, but not with the same territorial expanse as the Austrian Empire. A name can continue, but not necessarily all of the same area as before.