Aristotle's Metaphysics has perhaps the earliest mention of the Statue and the Clay (actually bronze in his example), but his hylomorphic theory sees no problem with the coincidence of material (ὕλη ) and the form (μορφή) of the statue.

The term “substance” (οὐσία) is used, if not in more, at least in four principal cases; for both the essence (εἶναι), and the universal (καθόλου) and the genus (γένος) are held to be the substance of the particular (ἑκάστου), and fourthly the substrate (ὑποκείμενον). The substrate is that of which the rest are predicated, while it is not itself predicated of anything else. Hence we must first determine its nature, for the primary substrate (ὑποκείμενον) is considered to be in the truest sense substance.

Aristotle clearly sees a statue as both its form/shape and its matter/clay.
Both matter and form and their combination are said to be substance (οὐσία).

Now in one sense we call the matter (ὕλη ) the substrate; in another, the shape (μορφή); and in a third, the combination of the two. By matter I mean, for instance, bronze; by shape, the arrangement of the form (τὸ σχῆμα τῆς ἰδέας); and by the combination of the two, the concrete thing: the statue (ἀνδριάς). Thus if the form is prior to the matter and more truly existent, by the same argument it will also be prior to the combination.

(Aristotle, Metaphysics, Book VII, § iii)

Aristotle also sees no problem with the body and soul of a person being combined in one substance (οὐσία), but a hundred or so years after Aristotle, the Academic Skeptics attacked the Stoics, saying Stoics were making single things into dual beings, two objects in the same place at the same time, but indistinguishable.

The two objects are just Plato's body and his peculiarly qualified individual (ἰδίος ποιὸν),
Aristotle would say they are his matter and his form.

. . . since the duality which they say belongs to each body is differentiated in a way unrecognizable by sense-perception. For if a peculiarly qualified thing like Plato is a body, and Plato's substance is a body, and there is no apparent difference between these in shape, colour, size and appearance, but both have equal weight and the same outline, by what definition and mark shall we distinguish them and say that now we are apprehending Plato himself, now the substance of Plato? For if there is some difference, let it be stated and demonstrated

(Anonymous Academic treatise, Oxyrhynchus Papyrus 3008 in Stoic Ontology, The Hellenistic Philosophers, A. Long and D. Sedley, p.167)

Perhaps the earliest statement of the classic puzzle of the Statue and the Clay was described by Mnesarchus of Athens, a Stoic philosopher who lived c. 160 - c. 85 BCE, as reported by the 5th century CE compiler of extracts from Greek authors, Joannes Stobaeus.

Mnesarchus' puzzle is the origin of the observation that the clay and the statue have different persistence conditions.

That what concerns the peculiarly qualified is not the same as what concerns the substance, Mnesarchus says is clear.

In this case, what goes in and out of existence is only what Aristotle called form (μορφή) or shape, the arrangement of the form (τὸ σχῆμα τῆς ἰδέας)

For things which are the same should have the same properties. For if, for the sake of argument, someone were to mould a horse, squash it, then make a dog, it would be reasonable for us on seeing this to say that this previously did not exist but now does exist. So what is said when it comes to the qualified thing is different.

(Stobaeus (I,177,21 - 179,17, in The Hellenistic Philosophers, Long and Sedley, v.1, p.168)

This is no longer Aristotle's ancient problem of the coexistence of body versus mind (or soul), or the Stoic problem of the material substrate (ὑποκείμενον) of a person versus the "peculiarly qualified individual" (ἰδίος ποιὸν), because modern metaphysics has become materialist, or naturalist, denying the dualism of a separate mental substance.

It is now common for many identity theorists to say that the whole of one object and the whole of another can occupy just the same place at just the same time. This is the problem of coinciding objects. Common sense says that two material objects cannot coincide.

In modern times, at least two puzzles are used to pose the problem of coinciding objects. One is the statue and the clay. The other is the ancient problem of Dion and Theon, in recent years described as Tibbles the Cat and a similar cat missing his tail.

But our puzzle maker asks us to focus on one and insist that the change has affected the status of only that one, usually claiming that the change has caused that one to cease to exist. This follows an ancient view that any change in material constitutes a change in identity. But the modern metaphysicist knows that all objects are always changing and that a change in identity may always preserve some information of an entity. The puzzle claims that an aspect of the object persists if the relative identity, or identity "in some respect" has not changed.

To create a paradox, we use two of our three axioms about identity,

Id1. Everything is identical to everything else in some respects.

Id2. Everything is different from everything else in some other respects.

We (in our minds) "pick out" one respect whose identity persists over time because of Id1 and a second respect which changes in time because of Id2.

We now have one object that both persists and does not persist (in different respects, of course), the very essence of a paradox. We call them different objects to create the puzzle.

In our case of the statue and the clay, Mnesarchus's original version assumes someone moulds a horse, then squashes it. We are asked to pick out the horse's shape or form. The act of squashing changes that shape into another relatively amorphous shape. The object changes its identity with respect to its shape. Mnesarchus said it would be reasonable to see this sequence of events as something coming into existence and then ceasing to exist. The most obvious thing changing is the horse shape that we name "statue."

By design, there is no change in the amount of clay, so the matter is identical over time with respect to the amount of clay. The clay persists.

We now claim to have seen a difference in persistence conditions. The object qua clay persists. The object qua statue goes in and out of existence.

But this is just a way of talking about what has happened because a human observer has "picked out" two different aspects of the one object. As the statue is being smashed beyond recognition, every part of the clay must move to a new position that accommodates the change in shape of the statue. There are changes in the clay with identical information to the change in the shape of the statue. These we ignore to set up the puzzle.

Notice that what we ignore is the identity of the statue and the clay. It is in fact the only true identity, the self-identity of any object with itself that is our third identity axiom.

Id3. Everything is identical to itself in all respects at each instant of time, but different in some respects from itself at any other time.

In more modern versions of the statue and clay puzzle, we can make a change in the matter, for example by breaking off an arm and replacing it with a new arm made of different material but restoring the shape. We now ignore the change in form, although it was obviously a drastic change until the restoration. For the paradox, we focus on the clay, making the absurd claim that the original clay has ceased to exist and new clay has come into existence. This is just sophistical talk. That part of the clay still in the statue still exists. So does the broken piece. It is just no longer a part of the statue.

There is a discontinuity when the arm is broken off and replaced, but after the replacement the newly repaired statue is still identical with itself at each instant, but following Id3 it is now a new self, different from its earlier, original self, with respect to the matter in the new arm.

All the paradoxes of coinciding objects are language games that ignore the fundamental identity of anything with itself.

In other puzzles, we are asked to make a change in only the matter (The Ship of Theseus or The Debtor's Paradox), or in both matter and form (The Growing Argument, The Problem of the Many, or Dion and Theon). A careful focus on the information always finds identical changes in both the matter and the form. The paradox focuses on one and ignores the other.