A Japanese city-state

Japan [...] has produced only one solitary city-state, Sakai.

Sakai, near Osaka, was an autonomous city run by merchant citizens which flourished during the Muromachi or Ashikaga shogunate, 1337-1573.

Ashikaga is the name of the clan. Muromachi comes from the Muromachi Street of Kyoto where the third shogun, Ashikaga Yoshimitsu, established his residence.

At the beginning of her authentically recorded history, Japan was a unitary empire, and in 1868 she became a unitary empire again. During the seven centuries ending in 1868 [from the establishment of the Kamakura shogunate, which lasted from 1185 to 1333, until the Meiji “restoration”] the political map of Japan was a mosaic of local states which had been held together during the latest two and a half of those centuries [Tokugawa shogunate, 1603-1868] under the hegemony of the most powerful of them, but, except for Sakai, these Japanese states had not been city-states. They had been feudal states, each of them ruled from a castle by a baron [daimyo] commanding a war-band of retainers [samurai].

The shoguns were military dictators. Kamakura was the city, thirty miles southwest of Tokyo, where the Kamakura shoguns were based. The most decentralised of the shogunates had been the Muromachi.

The establishment of the shogunate or bakufu at the end of the twelfth century saw the beginning of a de facto samurai control of Japan which lasted for seven hundred years, until the Meiji Restoration.

So three shogunates:

Kamakura (at Kamakura) 1185-1333
Muromachi or Ashikaga (at Kyoto) 1337-1573
Tokugawa (at Edo) 1603-1868

Tokugawa Ieyasu, who founded the Tokugawa shogunate at Edo, was from the Matsudaira clan of daimyos in Mikawa province and gained supremacy at the Battle of Sekigahara in 1600.

Toynbee seems to be implying that Edo was a feudal state, rather than merely the seat of the shogunate. No doubt it was: parts of Mikawa province (and others?) were administered directly by the bakufu. I assume that it also controlled directly some land around Edo.

Cities on the Move, OUP, 1970