Okay apparently I don't get a response cause you need way more information. So here is even more details although the basics I believe is already said:
Here's how I can replicate the problem. Very simple plain vanilla installation at GoDaddy
Create a subdomain other than www (my default URL now is http://what.mydomain.com)
Order a WordPress Application with all the defaults. This will install everything into the .\what folder.
Go to your WordPress site admin (aka Dashboard).
1. Left and bottom of menu, under Settings/General you don't even need to make any changes just click the button at the bottom Save Changes. Here I end up with the logon window for WP.
2. Left and Top of menu, under Posts click edit and click on the Hello World post to edit. On top of the edit windows you see Upload/Insert. Click any of the icons which should open a window for media upload. In this window click any of the other tabs and that's where I end up in the precise same logon window.
If I right click and view the source of the webpage, either for the media page or the login page, search for "what/" I find a line like this
<input type="hidden" id="_wpnonce" name="_wpnonce" value="15efbbacc6" /><input type="hidden" name="_wp_http_referer" value="/what/wp-admin/post.php" />
The relative path in value is definitely wrong and I have no clue how or why this would show up but, I'm most certain this is causing all the havoc. From what I know WP should not be aware of the path in the actual www-root.
Explained this to GoDaddy Support and got following response
Unfortunately, what you're describing is beyond the scope of our support and we would be unable to assist you. We apologize for any confusion in this matter. Please let us know if we can help any other way.
Regards, Tony P.
Online Support Customer
Meanwhile I searched some more and I found something I consider a work around.
Okay so I have a Windows host configuration where when I ordered the sub-domain a folder within my root directory was created and the WordPress installation was put into that folder. So far so good. Works perfect. My understanding now is, that we have a virtual directory and probably Host Header tie this together. Hence, for WordPress /what should be the root where my host, aka subdomain, http://what.mydomain.com resides. Furthermore, the configuration in WordPress is exactly like that. Nevertheless and for unknown reason, WordPress is apparently able to pull out the /what path from somewhere.
Now I've given a second thought to the configuration settings within WordPress where I can declare that the installation is in a different folder than the root. Surprisingly this is working now but quite frankly doesn't make any sense to me and I'm not sure if this cause way more problems down the road. I also see that links to wp-admin are prefixed with this path.
I hope this is enough input now, actually that is all that I know and can give. The actually website doesn't matter, I guess, cause I'd had to give y'all the Admin account to see what the problem is.
Any thoughts on this or comments what the correct configuration should be?