What's wrong with this MP4 file? Is the pixel aspect ratio wrong?
You are correct. The pixel aspect ratio is wrong. DV has a fixed horizontal resolution, but can display 16:9 frames by stretching out the pixels. The display resolution is set in metadata, so it is possible to make the file play back properly without re-encoding by changing the metadata.
The free, open-source tool ffmpeg can do it. This would be the command you would use:
ffmpeg -i infile.mp4 -c:copy -aspect 16/9 outfile.mp4