Similar technology to Chrome's Native Client Messaging in Firefox?
js-ctypes[1] is probably the closest alternative for Mozilla.
https://developer.mozilla.org/en-US/docs/Mozilla/js-ctypes
I have a C++ module that I compile as a binary executable for native-messaging or as a library for js-ctypes. The difference between the two is primarily that native-messaging calls a binary executable and performs stdin/stdout data exchange, and js-ctypes opens a static/shared library (via dlopen) and calls exposed methods of your library which can return compatible data types[2] and optionally call a passed JavaScript callback method.
[1] https://developer.mozilla.org/en-US/docs/Mozilla/js-ctypes
With new support for Mozilla 'Chrome' WebExtensions API seems like soon there will be a very similar technology to Chrome's Native Client Messaging.
At this time it's still on the 'List of APIs [mozilla] will likely support in the future'