Another clue. displaylink-driver-1.0.138.run is a collection of files which includes a canned script that (in my case unsuccessfully) invokes displaylink-installer.sh.
The .run file is created using a utility named makeself which is documented at
https://github.com/megastep/makeself
The makeself web page talks about an "embedded command" or "embedded script" which I think is displaylink-installer.sh.
Of possible interest, the web page contains a section beginning "Archives generated with Makeself 2.1 can be passed the following arguments:" for example --noexec --keep But note that the web page describes an option --verbose which is apparently not implemented by makeself (--confirm might be the equivalent of --verbose ? )