Only if the argument are distinct objects do the individual properties need to be considered to determine equality, which is a more expensive operation.
It might be a waste of machine cycles in the equality function not to take advantage of the discovery that the two arguments being compared are references to the same bignum.
It can still be referred to, and its other properties can be accessed via its external behaviour associated with the identity.
The identity provides a mechanism for referring to such parts of the object that are not exposed in the interface.
Identity allows the construction of a platonic ideal world, the ontology or conceptual model, that is often used as basis of object-oriented thinking.
The conceptual model describes the client side view to a domain, terminology or an API.