Their purpose is to be universally unique, presumably indefinitely. The practical choice is to align on a power of 2.
When UUIDs were invented, they were based on a MAC address plus a timestamp. MAC addresses are 48 bits, leaving 16 bits. Of these, between 1 and 3 are used up giving the variant code and another four bits are used giving the version code.
If they'd used 64 bits, there'd be between 8 and 11 bits left for a unique timestamp. Which is OK, but not a very strong guarantee for a "unique" scheme.
As it happens the length has made it possible to create multiple versions of UUIDs that are generated in different ways. UUID has been successful at separating the concerns of the structure of the ID from how it is generated. Most alternative schemes are effectively implementation-defined.