My question is what is the technology that allows that magic to occur?
The N80 is from 2000. By this time, EEPROM technology was already readily available at least in the sizes needed to store settings like those in a camera. EEPROM is a non-volatile memory (i.e. contents are retained even without power to the chip) that can be programmed many times over (typically thousands of times or more). It's relatively fast and durable. Data retention rates are typically well above a decade. It's likely that most cameras from this era leverage EEPROM technology to be effectively independent from any battery at all to store settings, frame count etc. So if you open up the N80, you may find that there's no auxiliary battery as you would find in cameras from 10-15 years earlier (e.g. Canon T90).
Yes most likely a high capacitance capacitor for the EPROM that hold essential settings.
Supercaps suitable for applications like these became available much later than EEPROM. I'm not aware of supercaps being used in film cameras for this purpose. That doesn't mean they don't exist, but it doesn't seem plausible to me for a variety of reasons.
Also, 'EPROM' is definitely NOT what's used here since that is the kind of memory that typically requires UV exposure to erase it. It had gone out of fashion several years before the Nikon N/F80 was introduced, and it's inherently not suitable for this particular application.