Switch statements are extensible in that you can add extra switch statements to your program without needing to go back and add a method to every class you coded, spread over a dozen different files. Its the old ExpressionProblem tradeoff.
All code is extensible if you have the source code and can recompile the whole thing then restart the program. I think pjmlp meant extensible in the "extensible at runtime" sense.
switch statements and method calls are kind of duals of each other. One makes it easy to add new classes but fixes the set of methods and the other makes it easy to add new methods but fixes the set of classes. It doesn't have to do with runtime.
Method calls don't need to be fixed either. Just because C++ stores virtual methods in a fixed-sized table doesn't mean Lua/Javascript/etc can't store them in hash tables. And a list of hooks is sort of like an extensible switch statement, but bare switch statements like you were describing obviously don't have that kind of runtime flexibility.
No, you can add new functions that switch over the different types without having control over the code. That way you don't have to add the same method to each of the classes.
