The primary function of debt is something called intertemporal smoothing. Essentially, it allows you to have the same consumption in every period even if you only have income in some periods. Maddison (2004) found that Song Dynasty China was 80% of the world GDP, so its obvious they had institutions to deal with this.

We have direct evidence that financial markets existed specifically for intertemporal smoothing. Ouyang Xiu (11th century) noted that farmers took debt in the summer and winter during the planting seasons, and paid the debt in the spring and autumn during the harvest seasons. He reasoned that the state ought to step in with low-interest loans, to insure those farmers whose winter crops failed, and they did, which led to the Evergreen Granaries, essentially a state agricultural credit bank (like the French Credit Agricole).

We also have direct evidence that the state was concerned with inflation, and had both monetary and fiscal policy mechanisms. Peng (1965) found that official price gathering missions were used to estimate three different indices of consumer prices based on composition of basket. We know the state used fiscal policy as a stabilising mechanism and had numerous secured and unsecured debt instruments, which Wang (1995) has fully compiled and broken down.

And finally we know that the Song bureaucrats cared about inflation through interest rate controls. The earliest treatise we have on interest rate management comes from Guanzi in 700BC. With the advent of paper currency, the state determined interest rate could serve as a floor rate. Over time, this rate stabilised progressively from 200% a year in 200AD to 12% by the Song Dynasty. We know that in periods of economic hardship particularly during floods, the state acted to lower interest rates directly, and we have numerous proposals (see Tang Jianjing (2016)) detailing policy recommendations on specific types of interest rates believed to have different monetary channels.

The perception that the West was the only country which practiced banking is influenced by an us versus them mindset that developed in Medieval Europe. At the time, 'us' were Christians, for whom debt and usury was frowned upon but hardly evil. And 'them' were Muslims, for many of whom debt was considered to be prohibited. The idea of applying juxtapositions unique to certain groups of foreigners onto all foreigners is a very common thing when you examine anything claimed to be uniquely Western.

1. Sort of? China's banking institutions, while developed by the late empire, were very different from European banking institutions. One reason is China's relatively weak system of laws during the empire, as China has always seemed to emphasize rule by men, not rule by law and thus banking, like ruling, relied upon close personal relationships. It was also common for business to lend between themselves, and this again, is due to the importance of personal relationships.

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It seems that the earliest form of paper money emerged specifically during the Song, but during the Tang, there were these letters of debt, sort of like a kind of proto-paper money, maybe akin to a check linked to a checking account? These currency forms were all tied to the idea of credit and debt in relation to the merchant economies of Imperial China. During the Tang, people started to realize that carrying around large amounts of heavy copper coins was a little cumbersome and so merchants and other sorts of folk who dealt with money on a commercial basis began to deposit liquid assets (cash) into so-called guifang or 'counting houses'. These same merchants would then be issued a letter of credit, and they would instead pay a small fee for the rental of this storage space. Ideally, people could then go about these guifang and withdraw cash or exchange letters of debt, but the system broke down when people began to turn them into gambling dens. Oops.

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Contemporaneously, there was the idea of feiqian or literally 'flying cash', which acted a bit like government bonds. They would be bought by merchants from the imperial treasury, and then when a need for cash arose, the government would then send the bonds over to whoever held the strips of cash, and possibly with interest? The exact mechanism is still unclear to me, and general understanding of the concept seems to be a little hazy in general from my research. Dr. Niv Horesh also notes that there's an 'increasingly sophisticated credit economy', without further exposition on the matter. But clearly we see here the emergence of at least some form of institutionalized banking, especially with the involvement of the central treasury.

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In the Song and beyond, this system simply expanded, and definitely by the Ming, there was most certainly the idea of money lending, and even utilizing interest in business transactions. I don't actually know if this system was regulated, and if interest rates were set and standardized though. Anyways, I believe that there was even passing mention of such conduct in "An Oil Peddler Wins the Queen of Flowers", a short story written by Feng Menglong as part of his anthology Stories to Awaken the World during the late Ming. I just re-read it, and there is indeed mention of money lending from one business to another.

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Sorry this answer isn't very expansive, I am not a subject matter expert on this material.

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Sources:

Horesh, Niv. Chinese Money in Global Contest: Historic Junctures Between 600 BCE and 2012. Stanford University Press. 2013.

Rogoff, Kenneth. The Curse of Cash: How Large-Denomination Bills Aid Crime and Tax Evasion and Constrain Monetary Policy. Princeton University Press. 2016.

Feng, Menglong. "The Oil-Peddler Wins the Queen of Flowers", Stories to Awaken the World. uh... Originally published in 1627, translated by Shuhui Yang and Yunqin Yang, published by the University of Washington press, 2009.