Dynamically creating multiple widgets from Mustache templates

How do I register multiple widgets with a single class definition?

You don’t. Modern class based Widget API in WP assumes that one class corresponds to one widget. Note that you can still have intermediary subclasses – (extend WP_Widget, then extend it once more) but what you register in the end is still one widget per class.

From your question it’s not very clear what your mustache templates are functionally and how varied they are.

If they are similar enough what you could do is create single widget and expose template list in interface to choose for each widget instance.