How can mainstream OO languages support strong binary operations by classes?

Question

Section 24.2 in Types and Programming Languages by Pierce compares ADT and existential objects,in terms of how well they support strong binary operations:

Other binary operations cannot be implemented without concrete, privileged access to the representations of both abstract values. For example, suppose we are implementing an abstraction representing sets of numbers. After scouring several algorithms textbooks, we choose a concrete representation of sets as labeled trees obeying some particular complex invariant. An efficient implementation of the union operation on two sets will need to view both of them concretely, as trees. However, we do not want to expose this concrete representation anywhere in the public interface to our set abstraction. So we will need to arrange for union to have privileged access to both of its arguments that is not available to ordinary client code—i.e., the union operation must live inside the abstraction boundary. We call such operations strong binary operations.

and says strong binary operations can't be expressed as methods of existential objects:

Strong binary operations, on the other hand, cannot be expressed as methods of objects in our model. We can express their types just as we did for weak binary methods above:

NatSet = {∃X, {state:X, methods: {empty:X, singleton:Nat→X,
member:X→Nat→Bool,
union:X→NatSet→X}}}

. But there is no satisfactory way to implement an object of this type: all we know about the second argument of the union operation is that it provides the operations of NatSet, but these do not give us any way to find out what its elements are so that we can compute the union.

but then says that mainstream OO languages provide support for strong binary operations by classes (instances of which are supposedly existential objects):

The classes in mainstream object-oriented languages like C++ and Java are designed to allow some forms of strong binary methods, and are actually best described as a kind of compromise between the pure objects and pure ADTs that we have seen in this chapter. In these languages, the type of an object is exactly the name of the class from which it was instantiated, and this type is considered distinct from the names of other classes, even if they provide exactly the same operations (cf. §19.3). That is, a given object type in these languages has a single implementation given by the corresponding class declaration. Moreover, subclasses in these languages can add instance variables only to those inherited from superclasses. These constraints mean that every object belonging to type C is guaranteed to have all the instance variables defined by the (unique) declaration of class C (and possibly some more). It now makes sense for a method of such an object to take another C as an argument and concretely access its instance variables, as long as it uses only instance variables defined by C. This permits strong binary operations such as set union to be defined as methods.

When a method in a class has an argument which is another object of the same class, how can the method "concretely access the instance variables" of the other object?

Does the book assume that the class makes the instance variables only privately accessible, and doesn't provide public accessible methods to access the instance variables? (I guess yes, because it seems to me that the third quote has the same situation as the second quote where it be impossible for union method on existential objects.)

Thanks.