Oh, boy.
Well, we're dealing with speculative fiction, so there are all kinds of empirical observations we're lacking. For example, there's no evidence that this kind of thing is at all possible, and the obstacles to its accomplishment probably have some bearing on the original question. But, what the hell...
I'm not convinced that Spock's mind can continue as Spock in McCoy's mind. Sure, we can write about the mind as if it's a discrete thing, separate from the body. But as far as we know, our minds are embodied; they exist as part of specific existence, in a specific time and place. Then you add another species' biochemistry, and the different experiences of McCoy. I'm not sure Spock-McCoy and Spock can be the same.
But let's assume Spock-McCoy is the old Spock, not long before death. McCoy keeps his (copied) mind alive for a little while, then he gets transferred into a cloned body, which more or less fits with his old body.
I suspect he'd be largely the same person, but somewhat changed for his experience. If anything, the knowledge that he's a clone, and the product of a disembodied mind copied into a cloned body, would change his outlook a little. As I suggested above, I also suspect he'd be changed by his newfound intimacy with McCoy.
In either case, is he exactly the same? No.