This is the first big misunderstanding in this whole affair.
That dialogue is
100% genuine (called from Apple API's). You can customize the text that appears ("blabla for Dropbox to work correctly") using older C-API's that are still fully allowed.
It is not a "custom Dropbox popup that can see your password".
This has been called out numerous times but unfortunately news articles can't keep up and that blog doesn't seem to want to edit the post (I'm guessing they're getting peak traffic from this story).
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EDIT: Here is the actual API's to achieve this:
https://developer.apple.com/library...f/#//apple_ref/c/func/AuthorizationCopyRights
Parameter: environment
"Data used when authorizing or preauthorizing rights. Not used in OS X v10.2 and earlier. In OS X v10.3 and later,
you can pass icon or prompt data to be used in the authentication dialog box"