That's called a "Jumplist" in programmer/technical terms, I built them using the Windows API when I develop my own applications.
They get validated depending on the state of the program, just like a right click (context menu) does. Are you sure that you're signed in, and have your application in the same "state" as he does when you compare?
For example, do you have any conversations open? If so, did he at the time you compared? Things like this will alter the categories and items in a jumplist, and you can even add to the jumplist yourself in some cases. Like with IE9.
You should only have that "Quit Skype" jumplist I believe if you're signed into an account when you have skype opened.
Application's that aren't running can still have services associated with the program running on your computer while the application isn't running. My guess is that a services associated with Skype becomes unresponsive when it has to close down on shutdown because it's corrupt or has a few errors since you last tried removing skype. Some entries from the previous skype install could have been left on your computer leaving behind all these conflicts.
Try doing this:
Press that windows logo key on your keyboard, and while holding it down, hit the "R" key.
In this Run prompt, type in "CMD" and hit Enter.
In this command prompt type in this command line and hit enter:
If you can post a screenshot of those running services and I might be able to point something out. My guess is that there's a service running on your computer associated with skype that causes this delay in shutdown, forcing you to force power off.
If you know how to copy and paste from cmd prompt then that's fine, otherwise a screenshot is fine enough. I use PowerShell, but I won't get you involved in that lol