Can't answer your question on a philosophical level but on an emotional/psychological level I would say that it's the same computer as long as it feels like the same computer to me, say by using the same desktop environment on the screen, and the same case, keyboard and mouse and if all parts have slowly been upgraded along the years not all at the same time as I would think that would create a kind of interruption in my relation to this PC.

To make a comparison, I bought my first SSD many, many years ago (to give you an idea, it was back when PATA still ruled). This SSD was tiny, it cost me an arm and a leg, and was not even that fast but even though I used it in the exact same computer I used with a classic HDD, the upgraded machine felt so incredibly much more responsive and so snappy that for me it was a new computer, and also a whole new experience.

Since that day, I've owned a few other machines, all SSD-based obviously, but never felt such a radical rupture while upgrading or even replacing a machine, no matter the CPU, ram or whatever else was new or better in it.