What you are really asking about is the science of thermometry, which is to say, the measurement of temperature itself. It is interesting to note that at least in the West, trying to "measure temperature" at all did not really occur until the 17th century. This is despite the fact that there were many contexts in which temperature mattered before then: cooking, alchemy, metallurgy, etc.

However I suspect that cooking is the most revealing and useful of these pre-thermometric approaches because most cooks today are non-thermometric to a large degree: unless you are using a tool by which you can very specifically set the temperature (like a modern oven), one tends to make things either hot or not, and use ones well-honed-over-time sense of heat to know when something is in the acceptable range. When I cook an egg, I set the burner to "in between maximum and off," and then I hover my hand above the pan to know when it is the right temperature. I don't even know, in a precise sense, what the right temperature is, thermometrically, but I know it when I feel it, because I have been cooking eggs for decades now (and am fairly good at it, unlike pretty much anything else I cook), and so have developed what is known as a tacit knowledge (embodied experience) that lets me easily judge what I need it to be.

One can imagine that prior to thermometry, one would see similar sorts of things elsewhere. For example, in metallurgy, one can define the type of oven one needs to achieve a specific result (say, the melting of iron), without knowing what the actual melting point is. Instructions can be similar operationalized: "wash the lead with quicksilver, then melt it," is an alchemical instruction that does not require thermometry, because it assumes that the desired end state ("melt it") is instructive-enough. (And someone who became proficient at alchemy would spend many, many years learning how to make such instructions real, just as an experienced cook knows what "beat until light and frothy" means, even if there is not a precise technical definition of "light and frothy.") And to your specific point, any alchemist worth their salt of tartar would know that different substances melted, boiled, and froze at different amounts of heat and cold — but they would not have some specific measurement to record for that, other than the fact of it (which is a measurement of its own, in a way: "heat until boiled," is just as precise an instruction as "heat until 212ºF"). Again, cooking is instructive: how many of us actually know the melting point of butter, precisely? (It's 90ºF, apparently, but I had to look it up.) It is not that important for the purpose of cooking to know this.

So this changes the perspective on the question quite a bit — why invent thermometry at all? That in Europe this happened in the 17th century is not very surprising, because this is when the study of the natural world took a more quantitative and observational turn, as part of what we sometimes call the Scientific Revolution. It was in this time that all sorts of new instruments and scales and metrics were developed for giving precision to observations across the gamut of natural experience, because the power of doing so was becoming evident.

It should be emphasized that thermometry involved quite a lot of work. It wasn't just Celsius and Fahrenheit showing up and saying, "here's how to do it." There were alternative approaches and stark disagreement on both what one was measuring (what is heat, anyway?) and how to measure it (should you use air, water, mercury, something else?).

The Celsius and Fahrenheit scales both rely on the so-called "two-point" system (pick two easily-measurable state changes — like boiling and freezing of water under constant pressure — and assume that temperature is a linear gradient between them), which itself was conceptually assailable (anyone who has tried to experiment with the knobs of a shower know that temperature doesn't feel like a linear gradient!), and also makes assumptions about how you measure it (does your measuring fluid expand linearly with linear changes in temperature? how would you know if it didn't?).

It is remarkable that thermometry got pretty much a lot of it "right," in the long run, given that the theories of heat that motivated much of the work (like phlogiston) were totally wrong.

Anyway, there is a lot more that could be said about the history of thermometry, and it presents quite a lot of interesting philosophical/historical questions about how to measure phenomena that one does not truly understand at a deep level. The definitive author on this topic is Hasok Chang (Inventing Temperature: Measurement and Scientific Progress, Oxford University Press, 2004), whose work, coincidentally enough, is what initially got me interested in the history of science.