Republican polities, in their broadest sense of an executive power being either elected or co-opted rather than taking its legitimacy essentially from dynastic virtue , can be found trough European history in various forms, not necessarily challenging aristocratic or royal power, during Antiquity and the Middle-Ages.Not being just pragmatical or ad hoc resolution, they were justified and supported by intellectual considerations (laws, constitutions, political philosophy, etc.)
What would be comparable to Mediterranean civic polities such as Greek, Carthaginian, Gaul or Roman ancient states or to medieval municipal republics, in Eastern Asian history?How did they led to a republican political philosophies from their own cultural background? How did these differed, or at the contrary were similar to what existed in Europe?
EDIT : Furthermore, were the Indian republican polities an important influence there?
Not too sure about the Indian question but I can try to answer the rest.
In Asia, there isn't that much of a republican tradition when you compare it to Europe. Just look at the main countries in Eastern Asia: China, Japan and Korea. China was under feudal rule until the 1911 Xinhai Revolution and it wasn't really until 1949 when the communists took over, that China had another stable central government under a republic. In Japan, the Emperor ruled (the power did fall to the Shogun but was retaken by the Emperor during the Bushin War and the following Meiji Restoration) until WWII was over, only then did the Emperor become a figurehead ruler. Korea was also under the rulership of an Emperor until early 20th century when the Japanese took over the country and put it under colonial rule. I don't know the history of Japan and Korea well enough to tell, but if you look at the history of China, most if not all uprisings resulted in new dynasties taking hold instead of seeing republics take hold, even if it was a peasant who rose against his lord.
If you compare it to Europe, that had republics for a few hundred years when it comes to more modern history and back then, had the ancient Greek/Roman republics, Asia doesn't really have a republican tradition even in history.
Now let's look into some of the shaping forces of Eastern Asian political philosophies. China no doubt had an important role in shaping the culture in east Asia. Confucian readings and academies were very big in all three countries. In the Analects, a collections of Confucius' teachings, there is a paragraph: "齐景公问政于孔子, 孔子对曰:'君君臣臣, 父父子子.' " The translation would be: Duke Jing of the Qi Dukedom asks Confucius how to rule, Confucius answers:"Kings be kings, courtiers be courtiers, fathers be fathers, sons be sons." It was some sort of a social contract, if kings rule like good kings and courtiers help their masters like honest courtiers, if fathers are good fathers to children and the children obey their parents, everything would go smoothly. This type of thought is often reflected in his teachings. Seeing as how Confucius was venerated in these countries, this may be another factor in the lack of a republican tradition.