There's kind of a two parts to this answer: why did "Caucasian" extend way outside 'people from the Caucasus region,' and why did "Caucasian" survive?

The term "Caucasian" was first used to refer to white people in general by Johann Friedrich Blumebach in the 1795 edition of On the Natural Variety of Mankind. Blumenbach was working from, well, skulls, and wrote one of the major works on craniometry. For reasons that are deeply unclear to me, possibly because I didn't spend multiple years measuring skulls, Blumenbach considered Caucasians to be the most beautiful and exemplary people within the white grouping (specifically dwelling on a Georgian woman's skull). He did also privilege the Caucasus as an origin point for humanity, believing that is where Noah's ark rested after the flood and thus the implicit center for the radiation of humans. For clarity, his other groupings were Mongolian, Malayan, Ethiopian, and American. This clustering is, to put it mildly, not sustainable by modern standards, being both wholly ignorant of genetics and vulnerable to more angles of critique than support, and had a pretty distressing belief that not only were Caucasians the original people, but that other people could be 'changed' back to being more Caucasian over time with environmental control.

(Interestingly by the standards of his time he was pretty progressive on race - he railed against the majority opinion that African people were less intelligent or artistic than other people, and no more violent or justifiable to enslave. Needless to say people kind of worked around his political opinions to get to the stuff they could use for scientific racism.)

So, what we have at this point is a single guy around 1800 who used terms that pushed for a link between skin tone and geography. The second question is, why has this usage proven durable and widespread? Why do other people use it? The obvious thing to point to is that Blumebach specifically was highly influential in his life: he was promoted rapidly from the completion of his dissertation, and had a pretty much continously rising career for the remainder of his long life, meaning that he taught an enormous number of students and was a member of a lot of powerful social and academic societies. Blumebach obviously did not get everything he wanted, but his ideas had a lot of reach.

A lot of the durability for the term of course comes from outside of academia, and this quote from the British Medical Journal provides some context:

The term has been largely abandoned by anthropologists, but it has been used until recently by immigration and prison officials in the United States of America and is still used in American medical publications.

To conclude, the term was made as part of an intellectual project that attempted to find patterns in skull shapes, geography, and skin tone, then developed a reputation for being a rigorous and strong term due to its inventor, and spread into a number of other contexts even as it died out within its home field.

Sources:

Painter, Nell Irvin. Yale University. "Why White People are Called Caucasian?" Yale University. 2003.

Freedman, B. J. "For debate... Caucasian". British Medical Journal. Routledge. 288 (March 1984): 696–98.

Unfortunately given its commonplace use, this term is derived from an early method of racist racial categorization by Johann Friedrich Blumenbach, in the 1795 edition of his book “On the Natural Variety of Mankind” that implicitly placed Europeans in a more “beautiful” racial class than any other group.

Blumenbach was a German psysiologist and naturalist who was active during the late eighteenth and early nineteenth century. To understand his racial system we’ll need a bit of background. The period of his activity was a time when science, as a way of thinking about and engaging with the world, was becoming central to how many people understood the universe. Concepts like what we now call the ‘scientific method’ were older, and it would be wrong to say this period was the foundation of modern scientific inquiry, but it was in the seventeenth and eighteenth centuries when there were massive cultural shifts towards beginning with scientific thinking as the main way of engaging with the world instead of just one among alternatives. So we can think about the goals of science to understand his project: the desire to discover absolute laws of nature, the idea that classifying everything we perceive into distinct categories is what lets us understand them, the use of theoretically objective data, and the attempt to sort everything based on such divisions.

When people talk about how racism is linked to science, this is what they mean. Dividing up all human beings into strict classifications, based on supposedly objective data (which was objective because the author doing the dividing said it was), and saying that that system of classification was what the world’s social structure should be based off of, was done by pretty much all of the major European scientific writers of this era. To them, strict racial categories—which were virtually always hierarchies—were just part of science, in the same way that categories like ‘mammal’ and ‘amphibian’ are to us today. The ‘problem’ was that no one could agree on what the ‘races’ were. Some European writers thought there were two or three races, others thought there were five or six, and the numbers eventually went as high as a few hundred!

On a side note, many genetic studies in recent decades have proved that there are no tangible genetic differences between the people involved in any racial system. I remember one professor I had in the past saying that the fact that all the racial systems that were proposed were so different and kept on having to be revised should have clued some of the racist writers in on this lack of actual genetic evidence, but what usually happened was that people would just keep proposing new systems, instead of realizing that biological race doesn’t exist. This is also where I need to note that race as a social category is important, absolutely does exist, and is deeply rooted in modern societies in part because so many people tried so hard to come up with scientific justifications for this social category.

On to Blumenbach and answering your question. So, he was living in a world with all sorts of racist classification systems, but he thought they were all wrong. Like others, he thought that he would be able to figure out what the ‘correct’ version of race was. And that’s what he tried to do. Here’s a summary of his system: He classified humanity into five “varieties”: “Caucasian,” “Mongolian,” “Ethiopian,” “American,” and “Malay.” For each group, Blumenbach assigned a series of characteristics, most of which are not worth repeating here, but the few differences in value he assigns to different groups are particularly relevant to unearthing the implicit racism of this text. He argued that they all possessed a different principle by which their societies were organized (on top of the already implied division of assuming each had distinct societies). In this conception, is only the “Caucasians” who are ruled by “custom,” the most reasoned social force that is adaptable and sensical. It is “Caucasians” who are central to this story. “Custom” is rule that changes the world intelligently and deliberately, and he explicitly wrote that it was only “Caucasians” who organize societies in such as way. The other main way he pointed out difference was in appearance. Then, he argued that these differences in governing principles and appearance were the result of whom each race was most directly descended from.

What makes the “Caucasians” so unique compared to all the other races Blumenbach made up is these two distinctions that set them apart: beauty and origin. His idea of beauty is the actual measure by which Blumenbach seems to divide racial categories, informing those peoples of which he values and disapproves. Yet he justifies this categorization by origin and thereby imagines himself doing the exact opposite of what he is really doing. That is, he arbitrarily identified groups of people based on arbitrary wants and arbitrary categories, and he then assigned a series of characteristics—mostly importantly origin—to those divisions of people, after which he inverted the imagined process, and argued that it was because of their origins that he categorized people in such a manner. Also during this time period, Greek art was viewed by many Europeans as the ideal form of art, which meant that someone who could claim to be descended from the Greeks would be more capable of art and beauty than anyone else. Writing that “Caucasians” have “that kind of appearance, which, according to our opinion of symmetry, we consider the most handsome and becoming,” he argued that Europeans were descended directly from the original people left in the Caucasus mountains, where Noah’s ark was supposed to have been left. That meant that Caucasians, who were the most original people because they were directly descended from the original people, were therefore more beautiful than any other race. So, once identifying Caucasians as a distinct group, he justifies them as inherently more beautiful—that is, more important—in comparison to all other groups, and he then does so with a theoretically objective method: symmetry. This is also the origin of the idea that facial symmetry is the source of beauty, by the way. Blumenbach has therefore chopped apart the human species into distinct races, argued that people descended from those in the Caucasus are the most beautiful, argued that it is Europeans who are descended from the Caucasus, and therefore that Europeans are more capable of good rule. Importantly, he managed to do so in a seemingly scientific manner, because it all rests on the supposedly scientific standards of facial symmetry and lineage. It’s roundabout because he says that Europeans are actually Caucasians because of their origins, but the way to prove this is because he says they are beautiful.

So why was this system of racial classification so influential compared to the hundreds out there? There is no way I could answer that fully, but I have some guesses. For one thing, Blumenbach explicitly condemned racism. Yes, I wrote that right: he believed that Europeans/“Caucasians” were more beautiful than all other ‘races’ and better governed, but not inherently superior. He actually criticized other racial classifications for saying that Europeans were superior to other races. In the same book that he proposed this system, he wrote that “we are with great probability right in referring all and singular as many varieties of man as are present known to one and the same species.” In practice, this meant that racist Europeans could use this theory without seeming racist. Important too was that his system rested on a supposedly scientific basis (facial symmetry+origins) instead of travel accounts and abstract theories, which meant that, in-line with the way science is supposed to work, it could be more trusted by more people. Its influence is still clear today whenever anyone talks about facial symmetry or uses the term ‘Caucasian’ for European people!

So, the long story short is that the term “Caucasian” comes from a racist classification system that says that Europeans are directly descended from the earliest human beings and are therefore more beautiful than all the other races it claims to identify, which you can tell because of their facial symmetry. None of which is true!

Sources:

Blumenbach quotes from Blumenbach, Johann Frederick. “On the Natural Variety of Mankind” in The Idea of Race. Edited by Robert Bernasconi and Tommy L. Lott. Indianapolis: Hackett, 2000.

Gould, Stephen J. “The Geometer of Race.” Discover (November 1994), 64-69.

Wheeler, Roxann. The Complexion of Race: Categories of Difference in Eighteenth-Century British Culture. University of Pennsylvania Press, 2000.

I’ll also note that there are a lot of really excellent books in the History of Science that explore topics like this, so a search on google or an online bookstore could almost certainly turn up some great readings if you want to know more. I basically just covered the background information off-the-cuff so if you’re more interested in general texts or specific instances of racial classification besides this one I can try to offer more readings too.

Edit: fixed a few typos.