The implementation technique outlined above is much like the one used by String and StringBuffer companion classes in the standard Java implementation, except that the mutable StringBuffer is not ...

For that to be possible, either you had to cast from mutable to immutable (which you shouldn't have) or the mutable string derives from the immutable string (which it shouldn't have).

Abstract classes and interfaces in Java serve fundamentally different purposes. Learn the differences between these Java language elements and how to use them in your programs.