Can "modality" mean "having modes?"

This quote from Jeff Raskin's book, The Humane Interface (courtesy of wikipedia) gives a good definition for "modal" in this context:

An human-machine interface is modal with respect to a given gesture when (1) the current state of the interface is not the user's locus of attention and (2) the interface will execute one among several different responses to the gesture, depending on the system's current state.

In your case, the "gesture" is "clicking on the widget's button", so the word "modal" is applicable here.

In general English usage the word "modality" means "having or related to modes" (for instance one might refer to "the modality of the music" about modal music), so your example would technically be correct. However, since the word "modality" has a specific different meaning within the context of user interface design, you might be advised to express yourself differently avoid confusion.

In computing a mode is "a way of operating or using a system," so if your widget has different ways of operating, which apparently it does, then it could be said to have a modal design.

In my experience, there are only two recognised uses of modality in computing:

• Modal Dialogs/Windows - These force interaction from the user; that is, the user must interact with the dialog (and ultimately dismiss it) in order to continue with other operation - http://en.wikipedia.org/wiki/Modal_window
• Human Computer Interaction (HCI) - Modality is the means of communication between the user and the computer (e.g. audio, visual, tactile)

Neither of these fit your scenario. That's not to say it is entirely wrong - model does mean 'pertaining to mode or form' - but at best, its use in this context might be unclear

Instead, I would say the button on your widget is multi-functional, context-sensitive or mode-sensitive/mode-dependent - I think mode-sensitive would be my preference.