Why do we use the word quarter to refer to an area of a town or city, usually one with a large population of a certain ethnicity, nationality, or cultural identity?
The word quarter has been associated with the number four for thousands of years, going back to the Proto-Indo-European root kwetwer, meaning four. A quarter can be a period of 15 minutes, an American coin worth 25 cents, or any other situation in which something is divided by four.
And this is probably where the word came to be used to refer to a subdivision of a town. It’s not clear if we can trace this usage back to one particularly city, but Jerusalem is a candidate, as its Old Town is divided into four quarters: Muslim, Jewish, Christian, and Armenian. One of the most famous quarters is the Latin Quarter in Paris, named after the Latin language, once used widely in the area, which features the Sorbonne university and many other higher-education institutions.
In the 16th century the word came to be used in a related meaning, with quarters first appearing in English to refer to military accommodation. This is also of course where we get the term headquarters, referring to the residence hosting the most important members of an organisation. This use of quarter is also probably the origin of the phrase to give quarter to someone, usually used as a negative order: Give no quarter! This now generally means to give mercy to an opposing soldier, but probably originally referred to giving shelter to an enemy in your quarters.
Of course it’s not surprising that the meaning of a word can drift slightly like that over time (though we’ve still kept the original meaning). In fact, in Middle English, it could be used to refer to any side or aspect of something. A coin might be referred to as six-quartered, meaning six-sided.
It’s interesting that quarter is the only word referring to a fraction that’s become so widely used. I think half is so specifically associated with 50% that it would be strange to use it any other way. And the words for all smaller fractions are probably too similar to their related numbers (six/sixth, ten/tenth) to be used with more general meanings. Quarter though, exists in a sweet spot between half and all the other fractions, clearly related to the number four, but looking and sounding different enough to be used in a more general sense.