There's tons of ways it can happen really. For starters in many cases a country's "global" name, in English or other languages, is dictated by the country itself and is just the native name of the country. For example Turkey recently demanded that it is referred to as Turkiye because they didn't like the fact that the name was the same as the bird. At least in all official diplomatic/academic/informational contexts, this has to be observed.

Some countries are named after their people, who had names as loosely defined ethnic groups long before modern day countries and borders were a thing (France/Britain/Germany for example).

Some times a country may have multiple different names in multiple different languages. It really depends where each language "got it from". Depending on various cultural and historical contexts a language may take a country's name and other words as loanwords from another language that introduced them to it. For example the Romans and Greeks were well travelled peoples, or held vast territories, and had writing and record keeping systems as well as languages that were spread far and wide due to their influence, so for many languages things like region names may be loaned from Latin or Greek. In the case of Japan they were a closed off nation and mostly came in contact with neighboring countries until European trade companies made their way there, so many words and place names in Japanese are loaned from English for example.

It's also affected by the intricacies of each different language. Some times names mean something in their native language and they're translated etymologically in other languages, so the end result sounds completely different but means the same, or the name is phonetically adapted to the other languages but due to differences in writing systems and spelling it may sound similar, but still different.

Basically, there's tons of different ways something like a country name may be adopted into a language, and the study of the origin of words themselves is tricky by itself since some words can very easily be traced back centuries while other have a very hazy and confusing history, so really there's no rule for coming up with country names in other languages, you have to examine it case by case, as in per language and per country name in said language.

In the case of Japan in English there's various theories, none concrete though. This article illustrates the issue a bit, while also providing a possible origin

>The origin of the name Japan is not certain, but researchers say it probably came from the Malayan ″Japung″ or the Chinese ″Riben,″ meaning roughly land of the rising sun.

Historians say the Japanese called their country Yamato in its early history, and they began using Nippon around the seventh century. Nippon and Nihon are used interchangeably as the country’s name.

In general the names of countries and regions in different languages is a product of the history of the world itself. War, Empires, Trade, have all impacted this throughout the ages.