Looks like there is both an ARM and x86 version according to the docs. Probably need two different binaries, but you still get cross-OS for each architecture.
% curl -O https://cosmo.zip/pub/cosmos/bin/basename
% Total % Received % Xferd Average Speed Time Time Time Current
Dload Upload Total Spent Left Speed
100 663k 100 663k 0 0 440k 0 0:00:01 0:00:01 --:--:-- 441k
% chmod +x basename
Kind of. It can be read as a single binary, some supported systems will do that. In others, the executable is first parsed as a shell script. There's definitely more to it than a single binary.
Reminds me of a cool tool I once used, uudecode.com, which was a DOS binary that only used 7-bit characters and could decode uuencoded (base64 predecessor) files. Was useful for getting attachments through e-mail in the face of all kinds of filters.