Edit: I figured it out: The solution is over a process substitution operator:

join <(echo "$var1") <(echo "$var2")

See also the comment by @notabot@piefed.social below, this StackOverflow comment, and the GNU documentation..

It's comparatively straightforward to use the content of one variable in join by using - to tell it to use standard input for that file:

echo $variable | join - anotherfile

However, is there a way to serve both input 'files' from variables, avoiding temporary files on the disk?

It seems like the easiest way would be via creating and mounting a temporary partition via tmpfs,

mount -t tmpfs -o size=50m tmpfs /mountpoint,

and just create temporary files in there. And afterwards clean things up.

So far I've also attempted here-documents, but apparently this too can only provide standard input, so that the other input still has to be served from a file.

Maybe one can also try doing it via named pipes (mkfifo), but I fear this could introduce lots of potentialities for errors.