In this discussion cheese refers specifically to hard yellow cheese
RaMBaM explains that according to Torah Law, cheese made by non-Jews is Kosher. It is Kosher, he explains, because it cannot be manufactured from non-Kosher milk. Even if cheese is made from a combination of Kosher and non-Kosher milk, the cheese is formed exclusively from the Kosher milk. When the milk solids separate, any non Kosher milk drains from the curd together with the whey that separates from the Kosher milk. [RaMBaM MaAchoLos A’SuRos 3:12]
RaMBaM is not concerned with, and does not even address, that such cheese is made with rennet from non-Kosher beasts because rennet, not being a food, is always Kosher. Accordingly, before our Sages prohibited it, Jews regularly ate the cheese manufactured by non-Jews, in spite of it being made with rennet from non-Kosher beasts.
[This is the reason that honey, the nectar collected from flowers but significantly altered by enzymes provided by the bee, is Kosher]
In the same vein, RaMBaM [MaAchoLos A’SuRos 9:15] rules that the milky contents in the stomach of a freshly slaughtered calf, if cooked with meat, will not make a prohibited Milk-Meat mixture. The reason for this is that this stuff from the stomach is not food. RaMBaM explains that even though it may be edible whilst still a liquid, is still not a food because it will inevitably progress to become inedible. [The Mechaber actually understands that RaMBaM permits cooking them together in the first instance]
ShA YD 87:9 & 10 first records RaMBaM's opinion, that meat and the stomach contents may be cooked together, followed by another opinion that prohibits cooking them if the stomach contents are still a liquid. This second opinion however, concedes that the cooked finished product is Kosher, and the ban only prohibits cooking them in the first instance.
Now all this directly indicates that the contents of the stomach and certainly the rennet itself, are never considered a food. Therefore, rennet cannot be deemed to be non-Kosher, which explains why [before the decree] we ate cheese manufactured by non-Jews.
What is rennet? It is an acid enzyme produced by the stomach lining and can be harnessed to manufacture cheese in four ways:
RaMBaM explains [MaAchAssuros 9:15] that our Sages did not ban cheese made with the contents of the non-Kosher stomach [method 1]; there is no justification for such a ban since it is not food. Rather they banned cheese manufactured by non-Jews when they started to use the actual non-Kosher meat, i.e. they used methods 2. [We will discuss method 3 later] However, since the motivation for this ban was to protect the Jewish community from social integration with the non-Jewish population, they banned all hard cheese no matter which method was used to manufacture it, and even banned cheese manufactured without rennet. Such cheeses are made from various tree saps, and are identifiable by their texture to be manufactured from plant based rennet.
What is actually required to avoid the decree and have the non-Jew's cheese Kosher, is not clear. Today we generally Pasken like the Shach who proposes that it requires Kosher participation in the actual manufacturing process.
In MaAchAssuros Perek 9:16, RaMBaM explains the difference between making cheese by immersing a Kosher or a non-Kosher stomach into a vat of milk. The Kosher stomach is Kosher, its meaty status is not prohibited but forms a prohibited combination if it imparts a discernible flavour to the mixture. However, the non-Kosher stomach is already a prohibited product. Therefore, although we may not make Kosher cheese with a Kosher stomach in the first instance [we use one of the other methods] if we did make cheese in this fashion it will be Kosher because there is no meaty taste in the cheese.
However, when using non-Kosher stomach, that which makes the cheese [the MaAmid - that which makes it "stand"] is a prohibited product and notwithstanding that its meat flavour is not discernible and it therefore ought to be Kosher, as with cheese made with Kosher stomach, our Sages decreed that its critical contribution in creating this new food product, is discernible and it will therefore be prohibited.
When the stomach is dehydrated, it becomes a non-food and may be used in the first instance to make cheese. Such cheese [method 3] are Kosher, provided we have taken the required steps to avoid the Sages ban, i.e. we have Jewish participation in the manufacturing.
Similarly, rennet even if extracted from non-Kosher stomach flesh, is a non-food and may be used to make Kosher cheese in the first instance.
This leads us to an obvious question: why is rennet not acceptable if it goes directly from the non-Kosher stomach into the milk [method 2] but is acceptable if it is extracted into a different medium, water for example, and later added to the milk?
It appears that our Sages intended from the outset to socially isolate the Jewish community from their non-Jewish neighbours. They wanted to shut down the common meeting point where we bought cheese from our non-Jewish neighbours, and sought an argument that would win acceptance amongst the people. The winning persuasive argument was, "They are making cheese with non-Kosher meat." which easily violated the common perception of what is Kosher. This simple but powerful and persuasive argument, empowered the decree to be applied across the entire spectrum of hard yellow cheeses.