[Update: The short answer is, if an application is the third thing on your taskbar, press WindowsKey+Shift+3 to open a new instance. Hooray!]
I have Chrome and Cygwin on my taskbar. Chrome's shortcut is Ctrl-Alt-C (as set through right clicking the icon and putting Ctrl-Alt-C into Chrome -> Properties -> Shortcut Key). Cygwin's shortcut is Ctrl-Alt-T.
When I press Ctrl-Alt-C, I get a new Chrome window. Great! It's as if I had shift-clicked on the Chrome icon.
When I press Ctrl-Alt-T, I get a Cygwin window the first time, but after that I just get focused on the Cygwin window, as if I had simply clicked on the Cygwin icon and not shift-clicked. Cygwin seems unable to have more than one instance open. I can still shift-click the icon to get more instances.
I've tried with different keys than Ctrl-Alt-T and got the same behavior. Strangely, I've twice managed to get it into a state (via just clearing and setting the shortcut key over and over) where a shortcut would open multiple instances -- but it was Ctrl-Alt-G both times, which doesn't make sense to my brain which has been trained to use Ctrl-Alt-T for years. Ctrl-Alt-G usually behaves just as poorly as Ctrl-Alt-T, except for the two times when magically it started behaving properly.
So I'm thinking this is a Windows 7 bug (which has existed since Windows XP at least), but I'm hoping someone knows something I don't!