Well, in practice you can't make your own non-enum types in stable Rust with this property, to unblock this Rust needs pattern types, (I wanted a path to do this a different way, but I was persuaded it's not a good idea). As that link explains the stdlib is doing this via a perma-unstable compiler use only attribute.
But yes, there are the NonZero integers, and you can make your own NonBlah integer using the "XOR trick" for a relatively tiny performance overhead, as well as you can make enums which is how the current CompactString works.
The link you gave mentions that Rust does this for other types, but in particular OwnedFd is often useful on Unix systems. Option<OwnedFd> has the same implementation as a C file descriptor, but the same ergonomics as a fancy high level data structure, that's the sort of optimisation we're here for.
Alas the Windows equivalent can't do this because different parts of Microsoft use all zeroes and -1 to mean different things, so both are potentially valid.