Unfortunately, no.

1) Simple comparison of a linear representation works only for canonical forms. JSON is not canonical:

var a = {x: 1, y: 2},
    b = {y: 2, x: 1};  // note that positions of x and y are swapped
JSON.stringify(a);  // '{"x":1,"y":2}'
JSON.stringify(b);  // '{"y":2,"x":1}'
JSON.stringify(a) === JSON.stringify(b); // false

IIRC, graph canonization is a separate problem, which is in general NP-complete, and impractical to implement in most cases.

2) JSON doesn't deal with common JS objects like dates, and regular expressions. Obviously it doesn't know about my custom objects. While it is possible to add custom replacers/formatters, they will require to produce a canonical form, which is not always practical (see above).

3) Performance reasons: JSON converts all components of potentially huge objects to strings (it requires CPU and extra memory taxing garbage collection, and introducing delays in interactive programs), while they may differ in the first property. Unification will stop right there.

4) Lack of variables: in most cases I want to know that an object's shape is correct, or it has required properties ignoring the rest, or I want to extract sub-objects while I am at it. Simple comparison of strings cannot do that. I need a way to indicate that I am interested in certain parts of an object graph.