Is every nonexistent thing identical to every other nonexistent thing? For instance, both the property of being Santa Claus and the property of being a unicorn have empty extension, so does that mean they are the same? Intuitively, we would say that Santa Claus and unicorns are both nonexistent, yet different. My question really is about whether properties with empty extension are indeed identical, that is they have the same intension as well. How does one rigorously define the intension of a property?
In a classical Frege-Russellian (inherited from a Kantian/Humean) second order view of existence, existence is strictly speaking not a first order property of things. Therefore there aren't any such things such that they do not exist: to be and to exist are one and the same (contra Meinongianism).
In most non-free, non-inclusive logics, including first order logic, having any predicate whatsoever entails existence (Fa ⊢ ∃xFx), including ones like being self-identical. Therefore, "nonexistent objects" lack any properties, including the property of being self-identical: you have to exist before having properties, and there just aren't any nonexistent things out there. Apparent reference to nonexistents is, in this view, a linguistic artifact which can be done away with once regimenting ordinary language sentences into their "logical form" (see Russell's descriptivist solution to the two puzzles of apparent reference to nonexistents & negative singular existentials)
So the answer, on this view, is that they're neither all the same nor all distinct: there just aren't any of them in the first place for them to have properties like distinctness and similarity.
If, by mentioning the concepts being both cointensive and coextensive, you're referring to "Santa" and "unicorn" as Quined Russellian descriptions replacing the constants with the quantifiers & variables a la Quine, e.g. "the property of being identical to Santa". While in the actual world, being Santa & being a unicorn will both have the extension of the empty set, in various possible worlds there are objects that satisfy Santa but not being a unicorn and vice versa, so they are in fact not cointensive. It is important to note that in following this approach, you're strictly speaking not talking about objects anymore, existent or otherwise.
Alternatively, you might be thinking of the problem of the semantic equivalence of all necessities and impossibilities in PWS approaches to natural language meaning: indeed, descriptions like "is identical to a square non-square" and "is a true contradiction" will all be false in just the same possible worlds, similarly for necessities which will be true in all the same worlds: these would be cases of cointension and if you follow the PWS mantra of "meaning is intension", we will approach a counterexample in the fact that "2+2=4" and "∀p p∨¬p" don't seem to have the same meaning.
It is worth noting that Meinongians, noneists, et. cetera will disagree here with respect to the background theory of existence. While the standard view is largely the consensus in analytic philosophy, I do not mean for this answer to implicitly presume its correctness. However, surveying the various answers to this question beyond the standard view is over the scope of this question, for which I would simply recommend surveying the vast literature on the metaphysics of existence & metaontology.
Meinong's jungle provides an insight into how non-existent things can be characterized by degrees of existence. Although Meinong's ontology allowed for the existence of non-existent things (which might be a run around of ur question), he does have some interesting categorizations of non-existent things like unicorns and mountains made of cake.