Kaeya, Dainsleif, the Alberich family are from here (including Chlothar Alberich/Eide, founder of the Abyss). Kaeya Alberich is part of the Alberich family. In Sumeru chapter part 6, Kaeya reveals that Khaenriah was located near Sumeru before its destruction.
Khaenriah is also referred to as Dahri in the Sumeru desert (see the Khvarena of Good and Evil quest with Sorush, also see Afratu's Dilemma).
As Eide (Chlothar) says, people of Khaenriah are cursed with the gift of immortality from the gods. Those who forsook the gods were welcomed to Khaenriah. It was a multi-ethnic nation. During the cataclysm, the pure-blooded Khaenrians were labeled as sinners, and the gods placed the curse of immortality on them. Those of other bloodlines were cursed with the wilderness and turned into monsters. This includes Caribert, Chlothar's illegitimate son that he had with a Mondstadt woman. As learned in the chasm, the curse is irreversible.
Khaenrians wear distinct clothes, have antipathy to the gods, and have special 4-pointed star shapes in their eyes.
According to Wanderer's stories, puppetry originated from Khaenriah. In real life, puppetry originated from Greece. So Khaenriah may be a reference to ancient Greece. Enkanomiya contains many references to Khaenriah people, many of whom have Greek-sounding names.
Khaenriah was allegedly destroyed in the cataclysm 500 years ago. In real life, Byzantine Greece was conquered by the Ottoman Turks around 500 years ago too. This may be overanalyzing, but it is just to play around.
The name Chlothar refers to a king of the Franks. He traces his ancestry to Pharamond or Theodemer. The Franks were a kingdom of Europe that fought against the Huns. Other parts of Khaenriah seem to refer to north European mythology. Altogether, Khaenriah may refer to ancient European civilization, including Greece, Rome, and the times before the Hunnic invasion.
There are some anachronisms, perhaps just due to Genshin's artistic decision to blend multiple time periods. One that seems to pass under the nose is the use of surnames. In China, surnames are almost taken for granted as many of the top surnames have histories spanning thousands of years. However, the rest of the world is not necessarily like that.
Alexander of Greece had no surname. The Roman emperors had no surname, including the greatest Roman conqueror Trajan. Europeans had no surnames for almost 1000 years after the fall of the Roman empire to the Huns. It was only until the middle ages around the 11th century that some surnames started appearing in Europe. And even after then during the Renaissance, some people still had mononyms. The kings and queens of Europe (even to this day) do not pass down surnames (see British royalty for example).
The reason this relates to Khaenriah and the Abyss is that Chlothar passes down his surname Alberich to his descendants, including Kaeya. However, the real historical Chlothar who ruled the Franks lived around the 500s and had no surname. He lived centuries before surnames became popular in Europe. So this is a kind of anachronism. It is not a big deal, but it shows that something we now take for granted (surnames) is actually a pretty recent invention in history, compared to other developments such as gunpowder, the printing press, etc.
In the Middle East, the term 'ibn' is used to denote the descendant of. It is a step up from pure mononyms and indicates the parents of a person. But it is still not quite like a surname.
Japan started using surnames during the Muromachi period (1300s-1500s), nearly 1000+ years after China. But even to this day, Japanese emperors do not necessarily have surnames (similar to British royalty). So if Inazuma is indeed based on medieval Japan, then perhaps the fact that Inazuma does have clans and family surname culture does make sense.
Could the development of surnames around the world be attributed to Chinese civilization? It cannot be a coincidence that Japan or Europe started developing surnames after the Gokturk era and around the time of the Mongol empire. The Gokturks and Mongols had deep connections to the Han Chinese, who had surname culture for thousands of years. It is possible the Turks and later Mongols spread this Chinese invention of surnames around the world all the way to Europe. It is similar to how the Chinese invention of gunpowder was spread from China to Europe.
Khaenriah was a very advanced civilization. Ancient Greece, by the time of Alexander, was also very advanced. Axiomatic reasoning used in mathematics was invented in Greece, in the book Euclid's elements. Classical geometric results were discovered in Greece. China has been using the same written number system since the Han dynasty. The concepts of zero and negative numbers were known in China since the Han dynasty too.
The modern Indian-Arabic positional number system, including zero, was invented around the 6th and 7th centuries during the Gupta empire. The Arab mathematician Al-Kindi introduced the number system to the Middle East. This Islamic scholars were well aware of the works of Euclid and other ancient Greek scientists. The Persian mathematician Al-Khwarizmi invented algebra, the concept of using unknowns to solve equations. This coincided with developments in algebra during the Song and Yuan dynasties.
Al-Khwarizmi is also the namesake of the word algorithm, which was loaned into English via Al-Andalus. In China, algorithms were known since the Han dynasty and after in books such as the Suanjing (算经), which detailed the Kushyar Gilani division algorithm. Incidentally, the modern Chinese word for algorithm is suanfa (算法).
The number system then spread west through the Middle East to Europe. Then Europe had a renaissance from the 15th to 16th centuries, taking the number system, algebra, and axiomatic reasoning of Euclid, to develop much of modern mathematics, including calculus (Newton). From there, modern math, physics, science exploded and advanced rapidly in a period of a few hundred years.
But the history of math can roughly be categorized in three eras: the ancient Greek era when Euclid's Elements appeared, the time between the Han and Yuan (Mongol) dynasties when the modern number system and algebra were invented and spread around the world, the time after the Mongol empire when Europe invented much of modern mathematics.
What math needed was a notation for numbers, a notation for solving equations (algebra), and then a system of reasoning (Euclid), and that provides the foundation of much of mathematics. Euclid was way ahead of his time. During Euclid's day, Greece did not have much of a positional number system (and seemed opposed to the idea of zero, see Zeno's paradoxes). To understand the need for a positional number system, there are different ways to write numbers. One can write out the word 'eight'. One can use something like the Roman system and write VIII (which is a bit of an upgrade as there is more organization). Or one can adopt a positional number system and write 8. Similarly, one can write 'eighty-five', LXXXV, or 85. Now how would one write 'eighty' using numbers? One needs the concept of zero to write 80. So something like numbers are not as trivial as they appear. It took nearly 1500 years for the ideas of zero and a positional number system to spread across Eurasia.
Another question one might ask is why do we even need numbers in the first place? Numbers help us calculate things a lot quicker. If we had 150 barrels of wheat from trade and grew 250 barrels of wheat in our farms, how many barrels would we have in total, and would it be enough to feed our country? Rather than counting each barrel one by one (using sticks or something) to compute the sum, we can write the amounts as abstract symbol 150 + 250. Then taking the sum becomes a simple algorithm: just sum the digits and carry over and we get 400. The idea of connecting quantities with written symbols in an efficient and organized system, something that we now take for granted, took a thousand years to become commonplace.
Euclid's idea was different but also profound: it was to use a logical system based on a few fundamental axioms to derive theorems via a sequence of permissible deductions / operations. This system of logic and axiomatic thinking is necessary or else one can derive anything from anything. It is almost a pity that during Euclid's time he could not take advantage of a positional number system or algebra and put that axiomatic thinking to greater use. This task would be done by Europeans 1500 years later, and the result is basically all of modern mathematics. But what Euclid was able to do in his Elements was deduce many geometric theorems without need of a number system or algebra. His idea of using axiomatic reasoning was 1500 years ahead of his time and is the same idea used by modern mathematicians when they create new theories.
The last piece of the puzzle is algebra, invented by Al-Khwarizmi. Algebra can be thought of as the next step of evolution after one has an efficient number system (positional number system with zero). It is almost like a kind of generalization of numbers to unknown values, quantities, and variables. Why would something like algebra be useful in real life?
Say you have 100 barrels of wheat that you want to split evenly among 20 houses in your kingdom. How would you do it? This involves a very simple algebraic equation 20x = 100. Solving for x gives x = 5, so you give 5 barrels to each house. Ok, but you could have also done 100/20=5 directly. So here is another example.
Meanwhile, many other world-changing things like paper, the compass, gunpowder (invented in China) were invented in that time span too. These other material inventions were also necessary for the development of massive empires that could harness the vast resources of a civilization to not just feed a population but develop math and science even more. The first thing many ancient empires were concerned about was how to protect their own empire and defeat their rival empires in war. To the rulers of the empire, things like developing the number zero or solving quadratic equations were almost secondplace and only useful if those things could help them gain an upperhand in war, for example by building more advanced weapons, catapults, cannons, and calculating the trajectories of projectiles against their enemies. Besides that the other application of mathematics was in calendars and astronomy, keeping track of the days and nights, using the stars as a guide when sailing and traveling. Such things would also require a kind of number system and some basic mathematics.
But for centuries, there was almost no need of something as advanced as calculus or quantum field theory unless it was to develop an insanely powerful weapon such as the atomic bomb or hydrogen bomb. 1000 years ago, gunpowder was invented in China, creating a huge revolution in warfare and empires. It is almost unimaginable that any civilization 1000 years ago would have been able to invent something like the atomic bomb. Such a thing would require a much deeper understanding of mathematics, which had not even developed to that point back then. The point is, while some things in math may not seem directly relevant or needed in life, the more one develops and advances mathematics, the more we can potentially push the boundaries of what we can create and advance our civilization. The invention of the atomic bomb, the computer, airplanes, spaceships, many of the modern electronics we use today relied on reaching a much deeper understanding of mathematics than what Euclid could have done 2000 years ago. Even something like a GPS, used profusely nowadays, depends on calculations involving Einstein's general relativity, which itself requires knowledge of differential geometry, calculus, tensors, etc.
A nation first focuses on becoming wealthy enough, growing the economy, to feed its people. Once they have a surplus of wealth, to advance the civilization and technology even more, it would be foolish not to invest in the development of math and science. To ignore the latest in science and technology is to fall behind as a nation compared to others. It is setting up your own decline. The basic research and developments in these fields are often the seed that can spark the next revolution. And many of the greatest and most successful empires in the past have also been the most technologically innovative.