As a general rule it is important to understand that due to the nature of translations and differing standards of veracity, I strongly discourage taking many numbers at face value as often times history was written by human beings who were just as sensitive to contemporary political pressures as people can be today.

With that being said, it's important to differentiate between the Ming Dynasty's "hundreds of thousands of soldiers" and contemporary Europe's armies. First, we must understand what the armies of the Ming Dynasty looked like.

As a general rule, the Ming Dynasty heavily decentralized its control of the military. The Ming fiscal state was a relatively weak one, as the primary forms of income came in the form of either agricultural based taxes, head taxes, or proceeds from a state monopoly on salt. The Ming routinely looked down on merchants and traders as all good Chinese bureaucrats did, so the vast amounts of trade and commerce that took place were left untaxed if not being outlawed by the government. It is also important to note that this lack of fiscal development also prevented the introduction of certain tools to temporarily increase government revenues, such as government debt and tax farming, to say nothing of joint-stock companies and financial corporations that we would see in Europe.

It is important to note this because this tell us some things about the Ming government's ability to maintain a standing army. As wealthy as China was, it was also very large and extensive. To fund a massive army in the hundreds of thousands that was sufficient to protect the borders would have been massively expensive. Even powerful and wealthy empires like the Habsburgs (who DID have financial tools and taxes on commerce) were strained by maintaining such a large army and were frequently forced to go to their creditors for financial support. At its peak, for instance, the Imperial Army of the HRE had likely no more than 150,000 men and that was in the middle of the 30 Years War.

So we rapidly come to the conclusion that quite frankly the Ming Dynasty, being financially undeveloped, could not support a large standing army in the European fashion. So how could the Ming possibly maintain its ability to defend its borders?

The answer lies in the devolution of government that the Ming implemented. Simply put, provinces were responsible for their own defense. Most people paid their taxes in two forms, currency and hard labor. Essentially, a few months out of the year, peasants were required to perform hard labor for some state mandated construction project. In lieu of this hard labor, soldiers were ones that instead served in the military. In practice, this meant that if there was some kind of incursion or otherwise, these farmer-soldiers would drop their plows, pick up their swords and spears, and go to war.

The Ming were clever enough to decree that local governors would need to maintain their own armies in this manner and pay for them out of their own taxes. Since central government taxes were collected by the provincial governors of which a tithe would be taken for the government, this meant that the central government could essentially pass on its military obligations to the local government. And by having soldiers that also fed themselves 9 months out of the year (meaning you didn't need to feed them), the government was able to save even more money. It was a brilliant plan that would make MBA students applaud. You could even write in your histories about how you could classify all the peasants on the borders as "professional soldiers" and boast about the might of the empire as your soldier colonists would be moved to an area, settle down, and occasionally fight down whoever was originally there in the first place.

This of course all became a problem when the Chinese faced an actual military opponent, of course. The best example to examine would be that of the Imjin Wars, where the Japanese warlord Toyotomi Hideyoshi launched a massive invasion of Korea, hoping to march all the way to Nanjing and take over the Chinese Empire. The Ming, concerned about this Japanese invasion, commanded the head of Liaoning province to dispatch an army to assist the Koreans in defending against the furious Japanese assault. This army numbered around 46,000 men, consisting of large amounts of cavalry and other troops experienced with fighting the Jurchen tribes on the frontier. After some initial success, they encountered their first real opposition in the form of a Japanese vanguard force of around 20,000 men. The Ming commander took his own vanguard of roughly the same size and attacked the Japanese forces, only to be quickly pressured and have his army put to rout by the Japanese infantry. Claimed casualties ranged from around 6,000 from the Ming troops to 120 killed and around 1500 wounded on the Japanese side. Simply put, the Chinese military arm, while sufficient at being cheap to maintain and useful in colonizing, settling, and pacifying regions, was less than adequate when it came to fighting a pitched battle against a semi-disciplined army, the relatively undisciplined serfs with likely ill-maintained and non-standardized equipment being little match against professional soldiers with years of experience in close quarter combat. The Ming also routinely had lack of success against the Jurchens in any sort of field battle (though they had some success when defending against a sieging Jurchen army). In a sense, the phrase that can best be used is "you get what you pay for."

Well, first things first, time wise the Ming corresponded mostly with Renaissance Europe. So the early comparisons would be with political units like England, France, Castile+Aragon, etc, and the tail end of the Ming coincided with the 30 Years War. The most straightforward answer to this question is probably just this: that the Ming was magnitudes bigger than the largest political units in Europe at the time. Early Ming census in the late 14th century counted about 60 million people. Later census had a relatively flat population growth, but it was well known that people were dodging the census (to avoid taxes) and real numbers were probably much higher. Estimates for the Ming population in the late 16th century is somewhere around 150 million to maybe even 200 million. In comparison, my understanding is that France in the 15-16th centuries had a population figure around 20 million or so. You can see how numbers by itself was a major player in how big an army you could field. The Ming was a state that was simply a lot bigger than anything else in existence at the time in Europe in terms of population and being able to field larger armies would be default given the disparity.

Then you had the infrastructure of the military. The Ming founder instituted a system that forced military households to each contribute a soldier to the state. This soldier would be expected to go to the frontiers, settle down in military colonies, and serve as an able bodied male ready to fight if need be. These were all soldiers, at least in a technical sense of the word. This obligation was hereditary, so if you were in a military family, your family always had an obligation to supply this soldier - regardless of how large this family became. Tracking this was a nightmare, and different regions of the country evolved different strategies to do it, but simply put we can trace the rise of large, corporate lineages in the SE part of the country partly to this system that required people to keep track of family members. The military thus recruited was not very high quality, with desertion, low morale, etc being endemic. Nevertheless, the system persisted, so theoretically the number of soldiers in the Ming was always very high, even if in practice their usefulness was at least sometimes questionable (getting an emperor captured, unable to handle pirates on the coast in the 16th century, etc). These garrisons existed throughout the country, and theoretically was supposed to be a low cost way of supplying soldiers for the state. This it did, but their lack of usefulness meant that it didn't exactly do the job required of them.

As you can see, the field armies size was a bit theoretical. At the Tumu Crisis where an emperor got captured during an imperial expedition to fight the Oirat Mongols, the Ming army was supposed to be 500,000 strong. There's the traditional bluff factor which inflated the count of people who were mustered to some Big Round Number, but even if you discount the army by a factor of 5 it's still a large army of maybe 100k soldiers. The supplies, however, did not keep up, and one of the reasons the army lost so badly was it was low on food and got too far ahead, eventually getting cut off from water supplies as well for a few days. Between poor leadership, poor readiness, and poor logistics, the army was no match for the much smaller but mobile Oirats. The emperor got captured, and the civil officials in Beijing, instead of paying a ransom to get the emperor back, just gave the throne to a new emperor instead.

Eventually, the military household system became unwieldy enough so that the obligation to supply a soldier got converted a cash levy and military conscription of this type was abandoned sometime in the 16th century. This reform was part of a series of changes in the fiscal arrangements of the Ming state that converted a lot of labour/goods obligations to cash taxes during the 16th century, probably fueled by the influx of silver from the New World that enabled a more cash based economy. It evolved out of families' strategies to deal with the system - nobody wanted to serve, but they had to come up with someone, so what to do? You pay someone else to do it (and maybe adopt them into your family if you must to keep up the pretense that they're a family member). The transition to a cash levy was just a formalization of something that was already happening anyway. Formalizing the payment to cash made life easier for everyone. Late Ming military formations were mostly recruited men who were paid for the job, even if officers often still came from military households with a tradition to serve. They were more expensive but when led properly were formidable fighting forces. Qi Jiguang, who was a pioneer of using these mercenaries, was able to suppress the piracy on the coast using a raised army of mercenaries. Yuan Chonghuan, who guarded the NE border against the rapidly growing Jurchen state, was quite effective in a series of defensive campaigns against the Jurchens. Yuan was executed by the emperor under false charges of treason, with the real reason probably (we do not know for certain) due to him controlling a large, what was essentially a private army at that point, even though the money paying for the army came from the central state. His army was said to have numbered somewhere around 150k at its height.

On military households and their problems/strategies, I highly recommend Michael Sonzyi's new book, The Art of Being Governed: Everyday Politics in Late Imperial China, Princeton UP, 2017.