The only "man-made" birds are domesticated species, which at some point in time trace their ancestry back to some wild species. They are man-made in that they don't exist in nature in their current form. They are not man-made in the way that implies they never existed in the wild. They are selectively-bred, domesticated animals. This is true for finches as it is for cats (descended from - and same species as - the African wildcat), the domestic dog (a subspecies of gray wolf), and all other domestic species.
There are some domestic animals whose natural ancestors are believed to be extinct, such as the guinea pig and the cow. In these species, the population taken captive by human beings and selectively bred has survived, while the ancestral population living in the wild has died out.
The domestic zebra finch is the same species as the wild one, simply bred into larger sizes and novel color morphs. The domestic canary is the same species as its counterpart on the Canary islands, but it has been domesticated longer and the captive population has been more selectively bred into various races, colours, and sizes, along with a general selection for birds that aren't afraid of people and readily breed in small enclosures.
The society finch is a domesticated subspecies of the white-rumped munia,
Lonchura striata. It originated as a cage bird in China, sometime prior to the 1600's, though the exact time of its domestication is not certain because bird keeping fell dramatically out of favor in the country during the 1800's, where the political regime in power discouraged and even banned the keeping of caged birds. It is believed that during this period, a lot of recorded history on the species was lost. Its origins are likely at least a few hundred years earlier, as I recall reading that some examples of Chinese artwork from close to 1000 years ago depicts white and pied caged birds similar to the society finch.
The society finch is often cited as originating in Japan. This is false; the Japanese did not obtain the species until the 1600's, though they certainly made contribution in producing new varieties. The original Chinese society finch was apparently only found in chocolate pied coloration, with the fawn variations as well as crested and frills mutations being produced by Japanese breeders over the following centuries. It is from this Japanese stock that all birds in the west - Europe and America - originate, thus the reason they are likely cited as coming out of Japan in the first place. While it's unlikely the Japanese would bring back plain wild munias from China in their search for novelty to bring back to their homeland, it was much more likely Chinese people hundreds of years before would have captured and caged them, since they were a common, local garden bird there which would have been easily accessible even to the poor common man, lending further credence to the Chinese origin of the society finch. Like the canary, these peasants and villagers would have caught the wild munias and then selectively bred only those which accepted captive conditions, and later to propagate unusual color mutations Occasional mutants with white spots would have been especially valued for their novelty, and it was over centuries of this methodical selective breeding they eventually produced the finch we know today, and which would later reach Japan through overseas trade routes.
Not only is the society finch so often cited as being ancestrally Japanese, but even more irksome is the continued parroting that it somehow originates as a hybrid species. This is
false, and the claim seems to have originated in a book of birdkeeping by a man named Dr Arthur Butler's, titled
Foreign Finches in Captivity, published in 1894. In this writing, Dr. Butler suspected the species to be of white-rumped munia and silverbill origin, but at this time this was nothing more than conjecture. It was later disproven. The first study to officially determine its origins involved comparisons of the skins of hundreds of self society finches and the various wild races of white-rumped munia in 1957, finding the society finch to fit well within the physical parameters of this species and finding no reason to suspect it descended from any other type of finch (if you look at a self chocolate society and a wild white-rumped munia, you can still the they are effectively identical, both in appearance and mannerism - see the attached image below.)
In more recent years, the society finch's taxonomy as a subspecis of whte-rumped munia has been proven with genetic studies. The species' genes have been compared side by side with the Indian silverbill (the species normally mentioned as the other contributing species in the society's ancestry), among other members of the genus, and they are most closely related, of all other munias tested, to the white-rumped munia. Further, the society finch does not exhibit any silverbill-specific behaviors which would be expected if it carried significant silverbill genetics. White-rumped munias and society finches breed freely and there are no behavioral differences in their courtship behavior, as occur in other munia species. Their calls are identical - only the songs of the males differ, because captivity has largely removed any selection in the society for a fixed song and thus most individuals - at least of different family lines - sing differently. Even so, the white-rumped munia and society finch have the most similar song structure of all munias. The silverbill's song and their version of the "puffy dance", conversely, is quite different. While most munias can hybridize, there are normally behavioral differences that make it less likely than among birds of the same species (I suspect most hybrids occur either as a result of uneven numbers in a captive collection, where birds cannot find their own species and accept another instead, or by sneak-mating. I observed my own captive spice finches, who outnumbered available females of their breed, very regularly sneak in behind courting male societies and jump on the society hens as soon as they presented themselves; even though spice finches have a similar courtship dance to societies, the hens would never show interest in them to let them mate in any other way. Male silverbills (who are behaviorally less similar to societies than spice finches, and have even more different vocalizations and courtship behavior, acted exactly the same, and were ignored otherwise by society hens. While this is only anecdotal evidence, I would not expect this if societies were part silverbill.)
The third species sometimes cited as being part of the society's ancestry is something called a sharp-tailed finch. I have only seen this species mentioned in very old texts, and cannot find a distinct species called this today, leading me to suspect it simply referred to one of the various extant subspecies of the white-rumped munia common across India and China. To go back to the issue of the society's origin again for a minute, it makes even more sense the species originally came out of China because neither the white-rumped munia nor the Indian silverbill even occur naturally on Japan to be domesticated in the first place! They had to have been imported; it's very unlikely they'd have bothered to import the wild munias, as they were nothing special, but the already domesticated pied forms would likely have caught their interest much more.
While it's possible some hybridization occured
somewhere back in the society's history, it is not supported by any study to have contributed in any significant way to differentiate the society from the white-rumped munia. The old story of the species being of some exotic hybrid, artificial origin is just that - a story, which is not backed up by evidence. The only society finches today which are of significant hybrid ancestry are the Continental selfs and black-brown societies bred in Europe, which are recent crosses of the society with the black munia and white-headed munia, among possible others, and some possible though unverified spice finch ancestry in some chocolate selfs, bred in to improve the scaling pattern on their plumage. In any case, these are recent hybridization events occurring only in the last few decades and which don't have any significance to the ancestry of the breed as a whole.
So there, that is my opinion on the "man-made" bird debate.