One's identity is routinely associated with one's body, and most will readily make the claim that they are their body. Their body is the referent for their identity. I'm having a bit of a problem just looking at this construction grammatically. If I say, "I am my body", am I not saying that the first person singular verb to be is equivalent to the genitive of possession? Isn't this grammatically and logically incorrect? If not, why not? If so, then who are we apart from our bodies or our identities?
I want to suggest an answer with two related aspects.
First, in the discussion about personal identity, there is another quite common theory, beside the theory that one’s body constitutes one's identity. The other theory is that one's memories (including habits and character) constitute one's identity. A major consideration in support of the memory theory is that we seem to understand the possibility of a “transmigration of souls” that is that we ourselves lived in different bodies in the past. It is unimportant here whether transmigration of souls actually occurs. Just that it is a coherent possibility. The originator of the memory theory of identity was John Locke.
But though the same immaterial substance or soul does not alone, wherever it be, and in whatsoever state, make the same man; yet it is plain, consciousness, as far as ever it can be extended—should it be to ages past—unites existences and actions very remote in time into the same person, as well as it does the existences and actions of the immediately preceding moment: so that whatever has the consciousness of present and past actions, is the same person to whom they both belong. Had I the same consciousness that I saw the ark and Noah's flood, as that I saw an overflowing of the Thames last winter, or as that I write now, I could no more doubt that I who write this now, that saw the Thames overflowed last winter, and that viewed the flood at the general deluge, was the same self,—place that self in what substance you please. (An Essay Concerning Human Understanding Book 2 Ch 27)
Second, the very question about identity arises only for rational beings. So it also brings into focus this fact, that we are rational beings (although I won't say perfectly rational beings..). That is, beings that perceive, understand, reason, decide, and (sometimes) act on reasons. And there are various contexts, where we think of ourselves primarily as rational beings, and abstract from the fact that we are also material, embodied beings. It is in such contexts, I suggest, that we tend to think of our bodies as something that we have, rather than something that we are. I, a rational being, have this body. This is also the basis for the metaphysical dualism of body and mind of Descartes's, and of Plato's Before him. But we need not accept the metaphysical dualism here, just the dualism of aspects.
The above two aspects naturally combine. When we perceive ourselves as rational, thinking beings, we can relatively easily think of our bodies as ours, rather than who we are. And we also have then available the memory theory of identity, which allows us to imagine ourselves as potentially floating from one body to another.