We're in serious "Ship of Theseus" territory here. Personally, I think the moment you switch mainboards, you have a new computer (even if mostly consisting of 'old' parts).
Absolutely -- it's like the PC of Theseus. I'm mixed on the motherboard thing. There's at least one instance where I changed only the motherboard, and so that didn't really feel like a "new" PC.
I think it was AMD AM3 to AM3+, enabled because AMD retained CPU socket compatibility. I needed a new board for another system, so instead of buying a new "old" board, I bought one for my main PC that was still compatible with my existing CPU (Phenom II ?), then used the old board for the other system. I later upgraded to a faster, newer generation CPU.
Agreed, the motherboard is what the computer really is. Maybe if I knew more about ship(building) I would have a similar opinion on the Ship of Theseus.