Not always. The one-click installer your host provides is not based on the control panel, though it may be limited by the control panel.
The big three script libraries, Fantastico, Softaculous, and SimpleScripts, can be run on most popular control panels, so most hosting companies will use one of these.
Fortunately, regardless of the script library your host offers, the most commonly installed applications will be available.
However, if you have a preference, this may be one more thing to research when selecting your host.