Both dramas are set in historical times, Joseon Exorcist being joseon and Bulgasal being both Goryeo and Joseon. Zombie like monsters are in both dramas, and they attack people in violent ways with super strength. They're more like demons, and there are specific demons that can turn other people into the mindless creatures to attack their own. Both are very violent and have good battle scenes.
Although cancelled, Joseon Exorcist and Kingdom: Ashin of the North feel very similar. They both have zombie beings hunting humans, and a dark historical atmosphere set in the Joseon period. They're both of similar length too, and while Joseon exorcist was unfortunately cut short, it can still be watched as a "what if" 2 hour long special.
They are both set in the early Joseon period. Both have large scale battles with a lot of violence and blood and with multiple heirs to the throne. Both have some kind of ''intervals'' inbetween that are usually followed again by intense and action packed scenes , and also thriller in Joseon's Exorcist case. Joseon Exorcist takes place after Prince Bang Won ascends to the throne, whereas in My country he is still fighting for the throne, meaning that one could say that in My country:The New Age could be considered a prequel, to some extent.