So the origins of the Fach, as you might guess from the name, are in Germany, predominantly in German repertory opera in the 19th century.

Repertory opera, as you probably know, is a troupe of players that performs a wide array of operas, wherein a single cast will learn a number of operas, and alternate performing them throughout the year.

To the best of my knowledge and research, it was simply developed by these companies as a means of pigeon-holing singers' voices, so that they could be cast in a group of roles that were suited to them, and thereby you could make an entire cast that could handle a full Rossini/Mozart/Donizetti schedule, and another cast that could do Verdi/Wagner and group it together that way.

As far as it developing with vocal pedagogy, certainly a ludicrous amount of pedagogy has grown up with it, and alongside it, but it doesn't seem to have pedagogical origins.

As a "perfect" definition, Fach would be directly related to the size and weight of the vocal folds. The lighter the folds, the lighter and more flexible the voice, and the heavier the folds, the stronger, louder and less agile the voice. However, in reality, it never works out this cleanly.

As a basic list of Fachs you have:

Coloratura: very light, fast, flexible

Lyric: less flexible, more weight, better stamina and lyric passages

Dramatic: Very loud, tons of stamina and weight (often literally), and less flexibility

And then you have the in-betweens:

Leggero: In between coloratura and lyric

Spinto: In between lyric and dramatic

And then to confuse matters more, you have Fachs that, while technically are real Fachs, they are more of a description of the type of character being played than of the style of voice required. These include the Soubrette Soprano, the Comic Tenor and the Heroic Baritone. While, yes, these are a valid Fach, it's pretty rare for someone's voice to be so specialized that someone would be ONLY a Soubrette Soprano, and not be able to sing the majority of the coloratura, or even lyric repertoire.

Then you get the Dramatic Coloratura. By all accounts, if the Fach system was simply a matter of thickness of vocal folds, this Fach shouldn't exist, and yet it does, and there are plenty of examples of actual, real Dramatic Coloraturas. While most of these roles are performed by Dramatics who have some good fast notes, or Coloraturas who can sing loud, there do exist a few of the real deal. Mozart's Queen of the Night is a notorious example of a Dramatic Coloratura role, and one that is almost exclusively performed by coloraturas, because almost no one else can hit those damned F6s.

And then you have people like me; the Zwischenfach (meaning literally "between Fachs") who can easily and comfortably sing the Mezzo Soprano and Soprano repertoire. If truth be told, I can comfortably to the entire contralto rep too, but since it's like 2 songs, I don't mention it. Again, that shouldn't exist. People like Maria Callas shoudn't exist, and she happily trod from Coloratura to Dramatic with little difficulty, and great acclaim. And there a plenty of roles with ridiculous ranges like Vitellia from Mozart's Clemenza di Tito which requires a G3 and a D6 from the singer. The former being out of range for any soprano, and the latter being beyond the expected range of all mezzos, and even many sopranos.

So then you get the vocal pedagogy, which, while it still uses the Fach system, rarely pigeon holes any singer into exclusively singing one set of repertoire the way Kloiber and the German Repertory Theatre would have done. While there are singers who specialize in a single style, or even a single composer, it's more common for a singer to identify with a certain Fach, but sing roles outside of it, but that still fit the colour and the timbre of the voice. Sometimes the classifications were set by the composer, and sometimes later by the opera community. Sometimes roles don't fit nicely into one fach, and many times you'll find roles listed as a different fach in different documents.

There's also the matter of how voices have changed. Verdi Baritones, for example, are a very new thing. Verdi has baritones singing much higher than they normally would, typically in the top third of their range. So the baritones that are comfy with this will specialize, or the bari-tenors will take it on because they're happy in that range, or the Dramatics with great stamina will just go for it. So who is the role for? And, more importantly, what do audiences want to hear? And what did audiences want to hear 100 years ago, vs. now? Now you expect Queen of the Night to be a top coloratura, even though that's not what the role called for. Carmen is a Dramatic and so is Don Jose, but they're normally performed by lyric singers in many companies, because that's what audiences want. Does it change the role, or the singer, or neither?

...so I realize this answer is a whole lot of speculation, other than the origins bit, but it's hard to nail down someone that is so contentious, and inexact, with so many exceptions and so many people that disagree with the system entirely.