OH!
This file is backwards from some of the above descriptions, and I think it’s the quicker/easier thing to fix.
The stream has the correct aspect ratio (16:9) indicated.
The container has the incorrect aspect ratio (4:3).
This is easy to fix, either with ffmpeg:
ffmpeg -i input.mkv -acodec copy -vcodec copy -aspect 16/9 output.mkv
Or in MKVToolNix, using the Header Editor -
Video display unit → 3
Video display width → 16
Video display height → 9
