A PDF file on my Desktop shows up with the name and properties of my root drive

Solution 1:

In all likelihood this is a bug.

A "real" PDF, or a malicious file (which would probably better hidden or disguised), would not show for "Get Info" things like "Capacity" etc.

If under Finder > Preferences > General the checkbox is set to
"Show these items on the desktop: harddisks" it just is displaying the correct name for the internal drive: "Macintosh HD" but mixes up somehow the filetype/association.

That means the suspected explanation in the question is exactly backwards:
Somehow the root drive/volume "Macintosh HD" is reported as kind "PDF".

To confirm this scenario, open a Terminal, type file (note the trailing space!) and drag the offending icon onto the Terminal window.

You should now see the following output or something similar:

$ file / 
/: directory

If that is the case, the solution is quite simple:

Open again the Finder preferences in the general tab and uncheck the "show hard disks on desktop" option,

and restart the Finder with a method of your choice. That might include a logout, login cycle, or just type in Terminal:

$ killall Finder

Now, even if you you re-tick that checkbox, your hard disk should show up with the proper icon and filetype.

If these steps fail, deleting the Finder preferences, and then the caches might be the next steps.