Every single province in Kozan is governed by a daimyo in the same way the feudal domains in Japan were. However, the daimyo you see in Kozan are not the same ones in Japan and the US equivalent would be a state governor. Daimyo in Kozan are chosen through direct election on years that end in an even number. All provinces in Kozan do this, but two of them are not.
The provinces in question, being North/South Horudika and Kabayashi don’t elect their daimyo, but succession is passed hereditary like it did in Japan. This is because those ruling families were royal families of two former countries that occupied the Kozanese continent. The daimyo family of North/South Horudika was originally the monarch of the former communist Democratic Kingdom of Western Kozan and Kabayashi was the center-left People’s Kingdom of Central Kozan. When both monarchs dissolved into the Federal Kingdom of Eastern Kozan, that monarch allowed the two kingdoms to stay in power as monarchs but under the daimyo title.
There are some limitations for the daimyo families of those provinces. Some members can still be elected into Parliament (and be elected Prime Minister) as long as they are not in the line of succession. Same thing goes for marrying into the Kozanese Royal Family. There are certainly some backstories behind one particular daimyo family of North/South Horudika.
There are legends surrounding a youkai-born woman named Saori Takarazawa (appearance will eventually be unveiled). She was born roughly around the same time as Crown Princess Konata somewhere in 1983. Because she was born during the Kozan-Soviet conflict, King Soujirou would not be able to meet her until he returned to Horudika Canal following the conflict in 1988 as he had fled in exile. Crown Princess Konata continued to live in Horudika Canal after returning in 1988 until she moved to Kozankyo in exile at the age of 11 due to a conflict within the royal family.
This move for Konata was hard on her as she was a good friend of Saori. They were an inseparable duo – sitting under the stars at night to pulling each others’ hair to even sharing Pocky sticks with each other. When Konata was leaving for Kozankyo, they decided to give farewell gifts in exchange with Konata giving a royal hairband to Saori giving a silver locket. They have not seen each other in a long time, as Konata never returned to Horudika Canal except for a few occasions. Truth be told, she actually lived where her cousins lived.
Rumors in Kozankyo are circulating that Saori Takarazawa came to Kozankyo, for something unexpected and unprecedented. Only in the next major Drillimation project can you find out the truths of her daimyo family.