Just tried the upgrade to Sierra (I can downgrade if necessary!) so now using Mac OSX 10.12 and Audacity 2.1.3. I installed it from the DMG file, I think, back when I was using El Capitan.
I used to be able to assign Audacity as the default program for a certain type of file, i.e. wav or mp3. Although I can still do that, now when I double click a file it only goes as far as opening Audacity, it doesn’t actually open the file within Audacity like it used to.
Starting Audacity by double clicking on a file was broken in 2.1.3 in that Audacity didn’t load the file even though it started. This has been fixed for 2.2.0 so this problem will go away when that comes out.
I confirm that with Audacity 2.1.3 using the right-click > Get Info > Open with or just using right-click > Open With on the audio you do just get Audacity opening but failing to import the audio file.
However… testing on the latest alpha nightly for Mac this appears to have been fixed.
As a workaround you should be able to drag&drop an audio file onto an already open Audacity window (doesn’t work if you drag&drop onto the audacity.app icon in Finder>Applications - it should - it does work in 2.2.0 alpha test build) .