For instance why is Bitcoin shown as BTC, Ethereum as ETH, and Ripple as XRP? Does every new coin have to pick a 3 letter designator?
BTC is not really BTC, but is XBT, since international currencies always start with X.
3 letters are just standard for currencies. USD, EUR, JPN, CNY, XMR, XBT, XRP. Same as with countries USA, GER, FRA, SVN, POL, ....
There is a rule that no official country code will ever start with an X, so it's safe to use codes starting with an X for things that aren't national currencies.
https://en.wikipedia.org/wiki/ISO_4217The use of an initial letter "X" for these purposes is facilitated by the ISO 3166 rule that no official country code beginning with X will ever be assigned. Because of this rule ISO 4217 can use X codes without risk of clashing with a future country code.
Some things that aren't national currencies like gold and bitcoin are called X currencies. Their codes start with an X to indicate they aren't national currencies.
X currencies
In addition to codes for most active national currencies ISO 4217 provides codes for "supranational" currencies, procedural purposes, and several things which are "similar to" currencies:
Codes for the precious metals gold (XAU), silver (XAG), palladium (XPD), and platinum (XPT) are formed by prefixing the element's chemical symbol with the letter "X". These "currency units" are denominated as one troy ounce of the specified metal as opposed to "USD 1" or "EUR 1".