Aristotle’s notion of identity is initially relational or structural, and ultimately processual. I read his emphasis on concrete entities in the Categories as pragmatic rather than ontological, and as of a piece with his general methodological recommendations about starting from what is close to us. Aristotelian starting points normally end up being substantially revised or discarded in the actual carrying out of an inquiry; they are intended just to begin a discussion in an accessible way. In the Metaphysics and the biological works, a nuanced and multi-leveled notion of form is developed that also represents Aristotle’s best thought on what we call identity.
For Aristotle, the identity of a thing at a given moment is almost reducible to what is true of it at that moment, i.e., to its articulable properties. There is no ineffable but contentful remainder underpinning individuality, like a Scotist haeccitas would seem to be.
I say almost reducible because the relational or structural criteria for identity of “this” continuing thing may change substantially over time, either straightforwardly because the thing itself changed, or more subtly because something else changed in the context of evaluation. Aristotle shares Hume’s scruples about applying identity in the first case, and Hegel’s about applying it in the second. Like Hegel but unlike Hume, he is also concerned to articulate and progressively improve criteria for higher-order stability of conceptual articulation in the face of first-order flux. This in a sense re-vindicates most of what is behind the common-sense notion of persisting “things”, provided we avoid overly rigid notions of the identity of those things. (See also Substance; Things Said; Aristotelian Dialectic; Being, Existence.)