When I am writing a post there's a featured image box in the document settings. It has a dotted outline and looks like I can drag and drop an image onto it to set the featured image, but I can't.
I would like to be able to drag images onto this box to set the featured image for the post.

I agree with this. I've received feedback from clients before on this after the release of 5.0. Visually it doesn't match what the intended result is. In my opinion, click to upload is fine but it should be a standard button. Drag to upload is better but maybe it was considered and there's a reason it's not available yet. However, if you're using Gutenberg as your editor and not classic the JS lib for drag and drop uploads is already loaded.
Ref:
For reference, I've created a ticket on Trac. Gutenberg is apart from Core at the moment but soon (hopefully) will be a full fledged part of Core.
@BinaryMoon Thanks again for suggesting this feature to EditorsKit plugin. Drag and Drop featured image upload is now available starting version 1.13. Requires Gutenberg plugin though, the filter is only available in there. 馃憤 Here's the plugin link : https://wordpress.org/plugins/block-options/

This is awesome @phpbits. I woud love to see something like this in core!
Can we just add that to the plugin directly @phpbits
@youknowriad Sure! I guess I'm late :) @epiqueras already created the PR.
Most helpful comment
@BinaryMoon Thanks again for suggesting this feature to EditorsKit plugin. Drag and Drop featured image upload is now available starting version 1.13. Requires Gutenberg plugin though, the filter is only available in there. 馃憤 Here's the plugin link : https://wordpress.org/plugins/block-options/