Note that "corporations" in Hamilton's time were not generally businesses, but rather organizations such as schools and charities that were charted by the state or federal governments. This was essentially an inheritance from Britain, where incorporation was bestowed as a royal privilege to certain institutions - typically municipal governments - through a charter.

You don't start to see business organizations chartered as corporations until the 1790s - a period of rapid economic development in which banks, canal companies, and other infrastructure projects received public money and public charters in order to build what was then known as "internal improvements".

At this time, the corporation implied a limited set of legal rights - they had the right to hold property, to sue and be sued - but typically they were set up to fulfill a particular public function.

It wasn't until after the Civil War, specifically with the Slaughterhouse cases, that you begin to see a more expansive definition of the business corporation. Using the newly-introduced 14th Amendment, which guaranteed equal protection under the law for all citizens, lawyers argued that the state's ability to regulate and tax corporations was limited because they were legal persons, and therefore could be treated differently from other citizens.

This in turn led to the rise of the corporation as the dominant form of business organization from the 1880s onward - very different from the late 18th century, when the great majority of businesses were family-owned or partnerships.

On the subject of corporations in Hamilton's time, you might be interested in Andrew Schockett's Founding Corporate Power in Early National Philadelphia.

Corporations as artificial "persons" goes back to at least Roman law. "Persons" in this context means legal entities able to hold property, enter contracts, and sue and be sued. Note that plenty of humans throughout history have not been categorized as "persons," namely slaves but also sometimes women, and other groups of people.

In my research, I've found that roman law provides the best examples of corporate persons, largely stemming from law relating to the state and municipalities, but also to private entities like "collegia" (which had a public interest, such as religious associations) and business entities called "societas publicanorum", which held leases from the state for things like mining and such. Societas publicanorum are fascinating early examples of business corporations, complete with shareholders, limited liability, etc. (see Ulrike Malmendier, Roman Shares.)

However, the notion of artificial persons goes much earlier, as far back as recorded history. This is best seen in the legal situation of temples in the ancient near east, which held property, could enter contracts, and sue or be sued. Some iron age temple systems in the ancient near east also resembled corporations in that the prebendary system resembled corporate shareholding (see Cornelia Wunsch, Neo-Babylonian Entrepreneurs.)

Bottom line is that the idea that organizations could be "artificial" legal persons is ancient.

Your question invokes one very interesting example - the Catholic Church was declared to be a corporation (or at least that church property was owned by/as religious corporation) by Constantine in the 313 Edict of Milan.

As the Catholic Church continued to follow various aspects Roman law throughout late antiquity and the middle ages, the various units of the Church (churches, monasteries, etc.) were also held as corporations, though this was contentious as both "secular" rulers tried to laid claim to church property, and ecclesiastical officials (bishops, etc.) also attempted to lay personal claim to church property (by trying to make it inheritable by their heirs, selling it for personal gain, etc.). However, that Church property was held in corporation was upheld in numerous law books and codes (Liber Iudiciorum 5.1.3, Corpus Juris Civilis Novel 120.6, etc.) indicates that this issue continued to be settled in favor of the Church as corporation over individuals.

When the Corpus Juris Civilis was "rediscovered" in Italy in the middle ages by the glossators (legal scholars who tried to reinterpret this late Roman law to the current context), the laws within it regarding corporation were very influential in outlining how groups of people may hold property not "in common" but "in corporation" (thus, not personally liable). One of the glossators was so lucky as to become Pope, and he, Innocent IV, wrote the famous phrase that corporations are "persona ficta", that is, "artificial persons". The important point here is that this is the common root in western legal tradition where it is understood that members of a corporation are not liable for the actions of the corporation itself. I believe the original example used is: if a town wrongs a person, the person can collect from the town, but not from the members of town.

But the town is a public corporation, and you are particularly interested in private corporations. Note that that all official public entities (cities, counties, states, etc.) do have legal corporate existence - thus, you can sue your city, county, state, country, etc. for wrongs - they can hold property, they can enter contracts, and so on like any person could.

Private corporations start springing up in the late middle ages and throughout early modern history. Early on they all have a public interest attached to them - they are hospitals, schools, etc. An important point to make here is that all corporations must be chartered by the sovereign, and they did so only for public-interest organizations. This is true even for business-oriented corporations like the Staple and the Merchant Adventurers. These were basically a combination guild and colony - merchants who organized under English law abroad.

The earliest English private for-profit corporation was the Russia Company in 1555 - this was a joint-stock company that bears significant comparison to modern shareholding corporations of today. This was followed by a very famous corporation - The East India Company. This is the one that eventually conquered much of India. There were many in this time period (1550-1700s), and they had a corporate governance issues. The early ones were like an alliance of merchants who agreed to work together and share profits (sort of like the earlier staples and merchant adventurers), but this doesn't work well in a corporate structure, and that was one of the early legal issues that was worked out.

One final point to make here was alluded to earlier - all corporations must be chartered by the sovereign. This was true in colonial America and early US history, but in the 19th century this was relaxed as many states (eventually, all states) enacted laws allowing easy and even automatic chartering. But this is outside of my research area and I haven't delved into it. Nevertheless, this is why even small businesses/organizations today can quickly and easily turn their company into an incorporated entity. Still, I hope this answer broadly paints the picture of how the corporation has always been understood as an "artificial person."