Brandon argues here, that if 'this cat' has no tail, but 'that cat' did not have a tail, then the title sentence is false.
Let's assume that 'this' is a true demonstrative, made in the physical presence of the object we are pointing to. But for the same reason, 'that' cannot be, for Tibbles-of-yesterday is not present – literally not present – for us to point to. So the 'that' must be some sort of referring description, perhaps 'the cat we saw sitting on the front doormat of the Jones's house yesterday'.
But why would we ask whether that was the same cat as this? Perhaps we are wondering whether the Jones have two cats or not. Brandon argues that the cat we saw yesterday had a tail, but this cat (here) does not. Thus 'That was not the same cat as this is'. Ocham agrees. Of course, the cat here is not the same, in the sense of *qualitatively* the same, as the one we saw yesterday.
But Ocham happens to have seen the accident when a car ran over this cat's tail, and it fell off. Thus, argues Ocham, the Jones really have just one cat. The cat we saw yesterday, and the cat we see here now, are one and the same cat. They are 'numerically the same'. Ocham agrees that this cat is not qualitatively the same as that cat, but they are numerically the same (and so of course are not really 'they' at all).
And I wonder, would we say 'the car ran over this cat's tail' (pointing to this cat here), or 'the car ran over that cat's tail' (meaning the cat we saw yesterday).