It was something mentioned in the OP that I took up and ran with a bit. I guess the metaphor represents a certain type of argument or thought experiment where one looks at reducing something to its basic constituent components. One can try to apply it to notions of self and draw a parallel between St. Patrick’s Cathedral (and its bricks and mortar) and the self. Does St. Patrick’s Cathedral have a permanent immortal essence? Where is it when one removes all its bricks? Does the same apply to the self? Or what happens if one gradually replaces all of its bricks and other components over time with similar but different ones? Will it still be the same cathedral?
This is the basically the Ship of Theseus. Though, a more modern and funnier version of this comes to mind. There’s a British sitcom called Only Fools & Horses, which has a character called Trigger. This character had a broom, which he amazingly kept in continuous use for 20 years (of course, the reasons for this are funny and much the same as this Ship of Theseus conundrum 
):
Some ideas like this do crop up in the suttas. The following from SN35.246 comes to mind:
"Suppose, bhikkhus, there was a king or a royal minister who had never before heard the sound of a lute. He might hear the sound of a lute and say: ‘Good man, what is making this sound – so tantalizing, so lovely, so intoxicating, so entrancing, so enthralling?’ They would say to him: ‘Sire, it is a lute that is making this sound – so tantalizing, so lovely, so intoxicating, so entrancing, so enthralling.’ He would reply: ‘Go, man, bring me that lute.’
"They would bring him the lute and tell him: ‘Sire, this is that lute, the sound of which was so tantalizing, so lovely, so intoxicating, so entrancing, so enthralling.’ The king would say: ‘I’ve had enough with this lute, man. Bring me just that sound.’ The men would reply: ‘This lute, sire, consists of numerous components, of a great many components, and it gives off a sound when it is played upon with its numerous components; that is, in dependence on the parchment sounding board, the belly, the arm, the head, the strings, the plectrum, and the appropriate effort of the musician. So it is, sire, that this lute consisting of numerous components, of a great many components, gives off a sound when it is played upon with its numerous components.’
"The king would split the lute into ten or a hundred pieces, then he would reduce these to splinters. Having reduced them to splinters, he would burn them in a fire and reduce them to ashes, and he would winnow the ashes in a strong wind or let them be carried away by the swift current of a river. Then he would say: ‘A poor thing, indeed sir, is this so-called lute, as well as anything else called a lute. How the multitude are utterly heedless about it, utterly taken in by it!’
Though, in Buddism, while the atta is being denied, this is being done in a context where, I think, some other things are being positively asserted, i.e., the cathedral/being can be dismantled/dies but, by cause and effect (unless the causes are cut off), that will lead to the later reassembling of a new cathedral, which is somehow linked to the old (that makes a rather awkward metaphor but that’s more or less what’s being implied I think).