These aren't nicknames, these are the standard names of US currency. Pennies, nickels, dimes, quarters, and half-dollars (not super common though)

They're nicknames for those (Similar to how people refer to bills by whatever president is printed on it), they might be very popular nicknames which grants them the "common name" descriptor, but the official names are the boring " cents coin". People outside of your country have no obligation to know how you nickname your coins.

Also if someone pulled out 26 coins to pay for a meal they'd also have a very annoyed cashier at minimum

Well, that might be true now because most counties only have 5 different coins, but pre-decinalized currency in the UK had 11 coins, it only got to 26 coins in my example because I included 7 of those (Sorry for farthing, pennies and Guinea fans out there), most of which in small numbers that someone might be carrying around in their pocket individually. And my point was precisely that, it's such a complex system that you end up with dozens of coins with random values trying to mix and match them to get to the amount you want.

The point of this was more "Coins are a pain in the ass regardless of whether we're dealing with 100 or 240 as the base"

But they're not, like you realized with 26 coins of 7 different values you didn't even get to a whole pound, with a decimal system the closest you can get is 1 50¢, 1 25¢, 1 10¢, 2 5¢, 4 1¢ which is 9 coins, and like you can see the vast majority is a single coin because 2 of them would get you to the next coin already.