Article IV, Section 4 says this:
The United States shall guarantee to every State in this Union a Republican Form of Government
That would mean no state can be a kingdom.
PS in response to comments: This section does not mean the federal government is guaranteeing to the states that the federal government will be republican in form; rather the federal government is required to guarantee that the state governments will be republican in form. That is done when Congress looks over a proposed state constitution before admitting a proposed new state to the Union. That has not always been done, since in particular it was not done in Kentucky. (On February 4, 1791, Congress passed an act saying the district of Kentucky in the state of Virginia would be admitted as a new state, of course with the consent of the Virginia legislature that had been expressed in 1789, but would not be admitted until almost 16 more months passed. The politicians of Kentucky had requested the long delay so they could use the time to negotiate compromises on details of their state constitution, which hadn't been written yet. That was the first time Congress passed a law admitting a new state, but because of the delay, Kentucky became the 15th state rather than the 14th (Vermont is the 14th).)