"if I could fit a newer motherboard into my MacBook, would it be unstable"
No it would be stable.
"would it still recognize the Model ID (MacBook1,1) and such?"
No, it's like doing a brain transplant. The body would be the same ( Uhh, actually only 15% the same on later examination!) but the brain would know it was different. See because you changed the brain (CPU) it knows it's different (Core 2 Duo capable of running 64 bits) vrs the old brain (Core Duo limited to 32 bit).
So in order to change the logic board from the early Core Duo to the Late 2006 Core 2 Duo the following items would have to be changed (Not a complete list.)
Battery, RAM, RAM door, Lower case, logic board ( of course), heat sink, DC in board, fan, top case, display,.....Uh it would have been a shorter list to list what was the same. You could keep the back of the display case with the Apple logo on it but just about 85% of the rest of the computer would be new parts.
See, that was a big jump to go from the Core Duo to the Core 2 Duo and that happened in the course of 6 months or so.
So I totally understand your attachment to your MacBook. I picked up an old Bondi Blue iMac that was going for donation to a kindergarden. The first iMac ever one made and this older guy had had it for 7 years. He asked to have a few moments with his iMac and he sat with his arm around it for 5 minutes while I waited.
When those iMacs went into the kindergarden the kids kissed and hugged them. You don't see that kind of love with Dell!
I still have my MacIntosh SE 30 and my IIfx sitting in my closet. I've hauled them from Chicago to Idaho to Portland over the last 25 years. So yes I understand.
Especially with the early iMacs. There were mods back then replacing the CRT with and LCD and putting an Intel Mini in place of the lower case.