According to the JLS:
15.9.5 Anonymous Class Declarations An anonymous class declaration is automatically derived from a class instance creation expre
Well, it would be pretty useless to be able to subclass an anonymous class. The only spot where you would be able to refer to the anonymous class would be in the statement where it's defined (as your hypothetical pseudocode example shows). This means that the program would be guaranteed never to create any instances of the anonymous superclass—and that a smart compiler should be able to collapse the two definitions into one class.
More practically, when a class is final, compilers and VMs are free to inline its methods at the calling sites. So in any situation where it is naturally impossible to extend a given class, it makes sense to make such classes intrinsically final.