It is unclear whether the Ruler of Sofa King is also known as "the Sofa King".
No units (other than Duke Nozzle) are concretely identifiable as belonging to Sofa King have yet been seen. It is unclear why. Perhaps the troops have always been there, but have been in the masses of infantry that can't be identified as being from one side or the other. Perhaps they were a reserve force and thus not at the front of the column. Perhaps most of Sofa King's troops are not Men, but are siege engines, such as the towers or cloth golems. They could even be the Weiner-Rammers. Only a Foxmud unit was seen riding a weiner-rammer, but that was after the deaths of Ansom and Duke Nozzle, so, assuming allied units can stack together (which they seem to be able to do), he may have commandeered it, or both forces may have had access to that unit type.
Real World References
The name of the side and names of its nobles sound like expletives when said fast. "Sofa King" was a Saturday Night Live skit in 2006 (). It was then used as a joke on Aqua Teen Hunger Force, which was sampled by DangerDoom, etc.
It may also be a reference to the Ottoman Empire, an ottoman also being a type of furniture.