A rough beginning on an argument. Identity is a very difficult concept to pin down, and there is more than one account that can be given of it. One of the big disputes in this area is over relative identity. On a relative identity position, at least some cases of a and b being identical can only be properly understood if we make this relation of identity 'sortal-relative'; that is, a and b can't be identical, full stop, but only identical kinds of things. So, for instance, Mark Twain and Samuel Clemens are identical men, and if someone says that two things are really identical, we can reasonably ask, relative to what sort of thing?
Lots of people find relative identity weird, and I am sympathetic, but I always find that many of the arguments against it are just obviously bad. And I think the problem is that people attack its relativity; but this is not the right piont at which to attack the account. The reason is that even 'absolute' or 'classical' identity, as it is usually understood, exhibits relativity; and this is uncontroversial, in fact, although how to handle it is not.
Identity is usually described as a logical relation exhibiting the following features:
(1) It is symmetrical (if A is identical to B, B is identical to A).
(2) It is reflexive (everything is identical to itself).
(3) It is transitive (if A is identical to B, and B is identical to C, A is identical to C).
(4) It follows 'Leibniz's Law' (if A is identical to B, then if anything can be predicated of A, it can be predicated of B).
(1)-(3) make identity a kind of equivalence relation; there are lots and lots of different kinds of equivalence relation, so we need to add something else to talk about identity in particular. Leibniz's Law is the least controversial proposal for this. (Incidentally, Leibniz's Law is not found in Leibniz at all. Leibniz does say things that sound somewhat similar, and you can get Leibniz's Law from things Leibniz says, but only if you make logical assumptions Leibniz did not make, and, indeed, would probably have thought dubious.) What's important for our purposes is that Leibniz's Law is known not to apply under every circumstance. To take an elementary instance, Hesperus and Phosphorus both name the same star. So Hesperus and Phosphorus are identical. But Fred thinks that Hesperus rises in the evening and thinks that Phosphorus is a completely different star. The result is a violation of Leibniz's Law: A and B are identical, but something can be predicated of A ("thought by Fred to rise in the evening") that can't be predicated of B. If we were to take this as a sign that A and B were originally not identical at all, it turns out that if we were consistent in doing that sort of thing, it would mean that almost overwhelmingly most of the things we call 'identity' turn out not to be identity at all. If we deny that a predicate "thought by Fred to rise in the evening" is not right kind of predicate, we make identity relative to kinds of predicate. And if we reject Leibniz's Law altogether, we need something better to put in its place. The first option simply changes the boundaries of disputes; so we usually aren't talking about identity in the strict sense, but we still would be talking about something, and the same questions would arise under a different name. It's really just a purely verbal solution, not a real one. If we take the third option and reject Leibniz's Law, then one of the main reasons not to accept relative identity as a genuine account of identity disappears. And if we take the second option we already are conceding that there is some sort of relativity built into the logical relation of identity, namely, relativity to what might be called a modal domain. And, indeed, any account that makes use of Leibniz's Law must distinguish between the right kind of predicate and the wrong kind of predicate even to get off the ground. This is widely recognized; it is often formulated explicitly to do this: if A and B are identical, then if anything non-intensional is predicated of A, it is predicated of B. The 'non-intensional' here makes the Law trivially true for identity, because it essentially means in this context that the predicates that are allowed are those that won't lead to the Law being violated. The only way to identify non-intensional predicates is by taking predicates with identical extensions and showing that substituting them in and out doesn't change the truth values of statements; so to identify whether something is non-intensional or not, we already have to know what identity is. In any case, we can only take Leibniz's Law to apply if we aren't moving across modal lines (like the lines between different things that Fred can believe).
If classical identity is relative to a modal domain, however, the only difference between classical identity and relative identity, then, would be that the latter takes every sortal concept (every kind of thing) to be a distinct domain, whereas the former only does this for certain modal groups of predicates. In other words, the relative-identity theorist takes there to be intensional or modal barriers between every sortal concept, every kind-of-thing concept, and their opponents do not. But the existence of the relativizing barriers in the first place cannot reasonably be put in dispute; the only question is where they are.