The shogun only really became the de facto ruler of the country during the Tokugawa era, before that they would only actually rule when the person of the shogun was capable of (like Yoritomo and Yoshimitsu), it comes down to how people see the "cart and ox" relationship between office and power.
The Kamakura Shoguns (even the powerful Yoritomo) weren't neither de facto neither de jure rulers of the country, their centre of power were the the eastern countries (Togoku) beyond Mikawa province, whereas the Heian court retained power over the western countries (saigoku), this arrangement made Japan a diarchy, at first you could argue that the court was the senior partner of the equation, however even the de jure authority of the Shogun on shogunal lands would vanish as soon as its founder, Minamoto no Yoritomo, died, being hijacked by his Hojo in-laws, when the diarchal relationship reversed in the Jokyu war following the end of Yoritomo's line the following Shoguns wouldn't even be bushi (warriors) but court nobles and minor princes, in actually more like court diplomats or even hostages of the true powerholders, the Hojo.
But here comes the question, both the Emperor, his court and the Shoguns had legal claims to authority, why were they powerless? There we go to the "cart and ox" dillema, is the Emperor/Shogun powerful because of the title or the power that the individual Emperor/Shogun possess grant them the authority? The Heian court had the legal authority over the country (at least the west) and the Shogun in Kamakura had the military leadership of the warriors under his demesne (mostly Kanto), but neither found themselves in a position to actual enforce their legal authority, the Emperor had long lost his power over to powerful Court nobles, the Shogun were nothing but puppets of the Hojos, as the Hojos increased in influence and wealth they soon became the true leaders of Japan, eclipsing even the Heian court, in the end the powerplayers believed the Hojos to be the actual chiefs, hence why Shikken became the leaders of Japan, and Emperor, Kampaku and Shogun were delegated to cerimonial or just minor roles.
The Muromachi Shoguns were the first to try imposing countrywide military rule, and were the weakest to do so (arguably from the inception), the times the bakufu worked it was due the individual efforts of the sitting Shogun, like Ashikaga Yoshimitsu, coming not from legalistic mechanism but personal, military or economic influence, the collapse of power in the East due to conflicts between the Muromachi administration, the Kanto Kubo and the Uesugi plus the Hatakeyama succession crisis that lead to the Onin War proved too much for the bakufu to handle and it only went downhill, the shoguns became once again puppets of the leading clans in Kyoto, to the point a Shogun Yoshiteru was murdered by his own vassals for his attempt to restore order. The end of the Muromachi goverment came, ironically, after a attempt to restore its order, albeit symbolically in fashion similar to Kamakura, as Oda Nobunaga installed his own puppet shogun, Yoshiaki, in Kyoto, however the Shogun conspired with his then benefactor's enemies he was kicked out of Kyoto and lived the next decade as a landless military ruler living on the good will of Nobunaga's enemies, the title had become so hollow at that point neither Nobunaga nor his successor Hideyoshi hurried to name themselves Shogun (the latter completely ignored it for the higher prize of Kampaku).
The Shogunate as we know today would only come in the 1600s, Nobunaga's former vassal Tokugawa Ieyasu successfully eclipsed his fellow grandees of the Azuchi-Momoyama period in Sekigahara and for several reasons (lineage, diplomacy or just tradition) went for the now vacant title of Sei-i Taishogun, and used his court titles to fully leash the Imperial Court, in 1615 he finally got rid of the Toyotomi clan and no individual clan could openly challenge the Tokugawa, Ieyasu built a system with several mechanisms to ensure his house's success, alternate attendence, one-castle-one-domain, the three-tiered vassal relationship, social stratification, everything ensured the Tokugawa were at the top, as the Tokugawas were Shogun, the Shogun became de facto rulers of the country.
For something resembling the shogunate in China or elsewhere you first need the power of the de jure leader (in the Chinese case the Emperor) to countinously degradate and the power vaccum is filled by a powerful official that establishes a tradition of that office being the actual power holders. You make special case for Japan, but there are two prominent happenances in the West: The Roman Empire (the Imperator being the shogun for the Senate) and the UK (the PM becoming the de facto head of government through the 18th century), now you need to apply that to China.
The Chinese Emperors weren't always those powerful greater than life figures popular history makes us believe, Cao Cao was essentially Shogun by the end of the Han dynasty, after Xianfeng every Qing Emperor was a pupper of their regents, have it evolve in something institutional.
