I think those are the perfect logs. I’m unfamiliar with the Shield, but wrongly-stretched DVDs annoy me, so you have my interest. 
I don’t think Plex is doing anything wrong.
From a playback perspective, this looks how we would expect:
08-02 19:23:13.350 i: [Player] onDisplaySizeChanged(720x480 at 0x0)
08-02 19:23:13.352 i: [Player][DimensionsLayout] Resizing: 1440 x 1080
It’s stored as 720x480. It should be played back at 1.33:1, which is what 1440x1080 is. Those numbers look good.
But it’s clearly almost a square on screen.
Are there “black bars” encoded into the picture on the left and right sides? On the Mac, if you shrink the window down horizontally, can you get it to the left and right edges? Or does it start to squish vertically before you get there?
All NTSC DVDs are encoded as 720x480. That’s neither 4:3 nor 16:9. At playback they’re either stretched or squished to the appropriate ratio.
Incorrect horizontal squishing can happen when a 16:9 DVD is marked as 4:3 on disc. At playback it gets squished to 4:3 instead of stretched to 16:9.
This disc might have 4:3 content that was rendered into a 16:9 DVD, along with black borders on the sides. If it was then tagged as 4:3, it could be extra-squished, like this, at playback.
(Everything about that is insane but plausible. It shouldn’t be necessary to render video as 4:3-in-16:9 for DVD, but it’s not uncommon for Blu-ray. If it was rendered as 16:9, it should be labeled as 16:9 … but clearly it’s 4:3, so it’s easy to understand how that could happen. And a clever Blu-ray player or a press of “Zoom” and it would make it through QA. I’m making excuses for them.)
A quick solution would be to grab MKVToolNix
. We can use it to change the ratio from 4:3 to 16:9, without re-encoding, so it plays back correctly.
MKVToolNix Downloads – Matroska tools for Linux/Unix and Windows
Drop your video on it, and then select the video track.
I expect the current Display width/height to be 640 x 480.
If so, change the Display width to 848. Save.
If they aren’t, please report back!