Hello columbusfoundation,
Thanks for reaching out to the Apple Support Communities! I understand that you're having trouble saving .mp4 videos to your iPhone. Nice work isolating the issue to only occurring with the videos which were shot vertically.
The first thing I would do, is to restart the iPhone with these steps: Restart your iPhone. After restarting, if the issue persists, try saving to the Files app.
When you select Export on a file in Dropbox, you'll see an option for Save to Files. If you don't see the option, you may see one that says "Open In..." After selecting it, you should then see Save to Files. I just tested this with photos from my personal Dropbox to my iPhone XS Max which is running iOS 13.3 and found the menus change depending on if you're in your own Dropbox or a shared Dropbox folder.
Once you have the videos saved to Files, check out this article: Use the Files app on your iPhone, iPad, or iPod touch.
If you're working with files that are in your own Dropbox, you can also follow these steps to add Dropbox to the Files app: How to add Dropbox to the Files app on Your iPhone or iPad | Dropbox Help
Once you have Dropbox accessible from Files, you'll be able to tap on the Share icon, and select "Save to Files." From Files, see if you can open and play the video. If you can, you should be able to save it to your iPhone.
Let me know if this helps.
All the best.