TeXnicCenter allows you to add shortcuts to menu items only. Moreover, these shortcuts are required to be a "single-combined keypress event". Let me explain:
In your preference, the choice of Shift+$ is contained in Shift+$+$. As such, you can assign the former to execute the menu sequence Insert -> Formulas -> Formula. This will insert $<cursor>$. Repeating this (by keeping Shift pressed and just pressing $ again), TeXnicCenter will insert a similar formula, which produces $$<cursor>$$.
This is how you do it in TeXnicCenter:
Tools -> Customize -> Keyboard
- Under
Category, choose Insert
- Under
Commands, choose Embedded Formula
- In
Press New Shortcut Key, while holding Shift, press $
- Press the
Assign button

Some caveats or considerations with regards to this technique: Whenever you use Shift+$ you will receive $<cursor>$, even if you only want to type \$ (say). Depending on your usage, this may be undesirable. Also, the use of $$...$$ in LaTeX typesetting for displaymath has been succeeded by \[...\] (see the l2tabu documentation). You can set a separate keyboard shortcut for this (the menu item Insert -> Formula -> Formula produces
\[<cursor>\].
To obtain exactly your preference (similar keystrokes to yield a choice between \(<cursor>\) and \[<cursor>\]) is not possible. You will have to produce completely separate keystroke combinations to obtain these different replacements. Note, however, that you can use Ctrl and Alt as well. So, for example, you could use:
- Ctrl+Alt+4 for
$<cursor>$
- Ctrl+Alt+Shift+$ for
\[<cursor>\]
With the above description, it should be straight-forward to do this.