History and custom. A quarter of a dollar is a convenient value -- but it's eight times less value than when I was a child. (A quarter then had the purchasing power of two dollars now.) A twenty-cent coin would make about as much sense, but nobody wants to get rid of the quarter, even though five cents is a trivial amount at this point.
The US issued the quarter because it was linked to the old Spanish monetary system of 8 bits to the Milled dollar and a quarter at 2 bits (which is still used sometimes ) made sense more so then a double dime . Bring it in later just didn’t work it terms of paper money $20 likely worked better given the gold backing and that $20 was two eagles
The quarter was from the Spanish dollar, which was divided in 8 bits. The quarter was 2 bits, which was a slang term until that last 40 years or so. The quarter eagle or $2.50 gold coin was derived from the $10 gold or eagle. A $25 note was precluded when Congress authorized the $20 gold coin. It was far more consistent with the decimal system.
In terms of the 20 cent(or double dime which I think was a much better name) piece the other issues was simply continuing the quarter which people were more used too. the US would never make the mistake of replacing a form of currency without recalling that form of currency nope that was a one time mistake for sure