If I understand correctly, the way Inkscape extensions work is by passing the whole drawing plus command line options to an external program, (python or perl), having said program process the svg and replace the drawing with the output - all this upon user input.

I would like to do something slightly different, which is have a process work continuously continuously and make things happen. It could be - for example - sync the drawing with another application via http, or remote-control inkscape from another app, also via http.

In Blender this is achieved with modal operators or application handlers.

Is something similar doable in Inkscape?

I suppose the simplest way to is monitor the output of `Edit>Preferences>Input/Output>Autosave`

That is a time based autosave folder ( you could check for the latest file in there ).

However there is no mechanism ( that I know of ) for event driven action of the kind you describe internal to Inkscape. 

Another way - possibly unnecessarily complicated, is to use an external program to run a custom extension using a dbus command from the external program.

But not, no clever internal automation exists at this time.