Regular non-system users (including Admins) are given
Read and Execute permissions to this folder and while this can be changed for other folders, Windows will not let you modify this folder in any way.
You may own the folder, but Windows will not give you full control to the folder.
For comparison I have given a picture which you attached along with screenshots of permission settings of my desktop folder:
My first screenshot:
My second screenshot:
Though I recommend not deleting the folder, the only possible way to do it is when Windows has not booted. You either need a pre-boot utility (there are a few for modifying SAM registries but not any for your specific purpose) or a second OS. According to me, the 'Second OS' method is better, you can get a live CD distro of almost any Linux OS and use it to do anything on your system that Windows restricts.
You may also use Unlocker to try and get rid of the folder though I doubt that it will work. Also the people over at Win SysInternals have created Handles to close bound processes (also known as handles).
Disclaimer: I take no responsiblity for any damage or harm caused due
to using the above stated methods or following the above stated