Edit: That marking matches the
AT24C02C, which is a very simple EEPROM, no unique ID, no special memory locations, not OTP locable... looks like a normal EEPROM.
Might be interesting to try disconnecting WP and pulling it to GND, sounds too easy.
Was the fresh one ever connected to the machine?
Are you completely sure both had the same contents?
Was it autodetected by the Programmer?
Try manually choosing a larger i2c memory, ex. 32Kbit.
There're special EEPROM with two different addresses, ex. the M24256 has a second address, for a page that can be permanently locked.
The AT24CS02 also has a second address to read a unique ID.
The AT24CSW02X has both.
Try manually selecting any of these EEPROM types in the programmer, or manually set the EEPROM address, using 0xB0 instead the default 0xA0.
Have you ever switched between two partially used dongles, and did the machine show different remaining times?