There's no king in Noxus, if I recall correctly. The city is ruled by a military dictatorship (think the Communist Party minus Communist) which is headed by a council (?) of high placed generals. Katarina isn't therefore a -real- princess, most probably the correct label would be noble or aristocrat.
As far as I understand from his judgement, Swain is quite the megalomaniac type, since he'd have to overthrow the entire High Command to set up a monarchy.... Plus he's pretty much a copycat off Ashe's intentions >_>;