EDID is a little EEPROM inside virtually all modern monitors with a standardized format, which can be read over the display cable [1]. It essentially enables "plug and play" for monitors, as opposed to the bad old days when you needed either an .INF file to install in Windows or the supported sync ranges to type into your XFree86 configuration if you wanted anything above 640x480. It contains data like the vendor, model number, physical screen dimensions, and supported display modes. There is a field for a serial number; parent is proposing using this to identify individual monitors. However, errors in EDID are quite common, e.g. using incorrect units for the physical dimensions or leaving placeholder/OEM data for vendor and model number that don't match the finished product's advertised vendor or model number. I imagine there are a lot of monitors out there with bogus EDID serial numbers, since it isn't important for getting the display modes right.

[1] https://en.wikipedia.org/wiki/Extended_Display_Identificatio...