Identity of Initial Elements in a Category I don't know much (really, anything) about Category Theory or Structural Set Theory, but I happened across this blog post which piqued my interest. The author claims:

But in Structural Set Theories, like ETCS and SEAR, one cannot prove uniqueness of mathematical objects, and, in particular, the uniqueness of the empty set. Instead, an empty set is defined as an initial object in a certain category. (Lawvere (1964) gave the first categorical characterization of the set-theoretic universe.) And while one can prove that initial objects are (uniquely) isomorphic, one cannot prove that initial objects are identical: one cannot express that they are identical.
So---at least as I now understand it---in category theory, one can't
prove, for objects $X,Y$ in a category $C$, a uniqueness claim: $$\big(\operatorname{Initial}_C(X)\land\operatorname{Initial}_C(Y)\big)\to X=Y$$ And this is merely because one can't express identity of objects: $$X=Y\;.$$

Is this accurate? Does Category Theory (and Structural Set Theory) typically make do without an identity predicate between objects? Are there serious obstacles to introducing one? If so, what are they?
 A: No, this is false. A category first of all has a class of objects, and relative to a fixed formalization of a notion of set theory with classes, equality of two elements of a class is a perfectly well-defined notion. 
The insight of category theory is that even if one can talk about equality, in many situations it is better not to, and talking about isomorphism is better. 
A: Well, in category theory based on the "ordinary" set theory there are no obstacles of using identity predicate. But in category theory you normally use objects that are determined up to isomorphism and it's all that matters.
You have objects and morphisms. If there is an isomorphism between two objects then they are identical from your point of view: you can't distinguish them in terms of any of their properties that can be examined in your world: world of objects and morphisms.
I'll not comment on the structural set theory part of the question, since I don't have much knowledge about it. But I like the idea of different empty sets - for IT specialists it is for example very natural to distiguish between empty set of integers and empty set of reals (floats). And it was quite shocking for me when I first saw in my set theory course that it is possible that $\alpha \in \beta$ and $\alpha \subseteq \beta$ at the same time. Afaik, with type theory and  different empty sets you introduce some hierarchy and avoid such pathologies.
