In Swift, if I have a class with only type properties and methods (everything declared static) would that be considered a singleton object? If not, why?
This isn't specific to Swift at all.
You don't have an object, you don't have an instance of the class, therefore you don't have the singleton.
The essence of the singleton object is that it is the sole instance of a class. Exactly one instance. Not two, not three, not zero. One instance. You have zero instances, therefore you don't have a singleton object.