Yup. The DVD is incorrectly mastered, the MPEG2 headers say the stream is 4:3. The reason it works on your DVD player is likely that the VOB files says it's 16:9.
These are the steps you need to take to fix it:
1. Extract the video stream from the MKV with MKVextract (part of the
mkvtoolnix package).
2. Download
DVD Patcher.
3. Open DVD Patcher and open the MPEG2 steam that you obtained in part 1.
4. Once changed to 16:9 save the file.
5. Remux the video stream created in part 4 with the MKV file using MMG (downloaded with
mkvtoolnix), untick the video stream from the MKV file so you don't end up with a video file with 2 video streams.