That’s likely more about hardware than software, but one of these threads might be relevant:
The driver has a small capacitor to measure off-time, but it’s not enough power to keep the MCU running. The MCU itself also has some SRAM which decays slowly so it can hold values from one boot to the next if the off-time is fairly short. Both of these methods are used.
As for eeprom, it uses wear levelling. One byte is written per boot, levelling across 64 cells. The spec claims 100,000 write cycles, though independent tests suggest it usually gets over a million before seeing actual failures. So, in order to wear out the eeprom, you’d have to press the button 175 times per day for 100 years before it’d fall out of spec, and likely more like 2000 times per day in order to see an actual failure within one human life span.
Config data doesn’t use wear levelling, but it also doesn’t get written unless you enter config mode. The fastest way to wear out the eeprom would be if you left it in mode-group-select mode for about 3 years.