Aristotle thinks that "substance" (the union of form and matter) is primary being. This puts him between the nominalist (only physical particulars exist) and the Platonic realist (transcendental things exist, like math). He looks nominalist when we consider that on this view the first things to exist are the physical particulars: the nested categories extend from the individuals outward, whereas with Plato the universals are primary being, organizing matter into categories. But he looks Platonic when we consider numerical identity and personal identity.
One might say, "I have two identical pieces of chalk." Usually when we say such a thing we mean that we have two pieces of chalk with the same form: it's clear that we consider each piece to be metaphysically distinct because it is a different piece of matter. So in our ordinary talk about physical things we don't accept identity between distinct pieces of matter. My idea of Aristotle the man is that he was the type of person who would say, "Sure, that's how it goes, and so you don't need to make up any exotic properties like 'The property of being identical to oneself' or anything like that," and wouldn't mind that his own analysis of substance doesn't technically underwrite this practice. According to Aristotle, the existence of all the physical particulars is just a brute existential fact (substance is primary being). Many physical particulars are similar enough to others that we can name these categories. So we have a category called "humans," but we don't have one called "Socrateses," but if primary being were arranged differently we could have a world where humans were further sorted into Socrateses and Aristotles. They're all horses, and they're all made out of distinct matter. I think that if we put Socrates through a malfunctioning transporter and two of him came out the other end, Aristotle does not hold that they are neither the original one. I think he has to hold that they are both Socrates. For better or for worse, Aristotle's substance account countenances the metaphysical possibility of simultaneous exemplifications of one thing: to him that's just quotidian, mere taxonomy. Of course they'd be two different bodies and in fact this makes all the difference. But then we'd make up new names to designate the two Socrateses. Which brings me to my final thought today: Looking at Aristotle's position here, it starts to seem like some criterion of physical identity isn't so obviously wrong. Maybe a person is just identical to their physical body.