First of all, the FTDI chips themselves have no firmware. They are implemented using fixed function logic IIRC. Even if they did have firmware, it'd be on a mask ROM and wouldn't be changeable. What FTDI chips and their clones do have is a configuration EEPROM. On some chips it's internal, but they do support external EEPROM too. That's where the VID, PID and USB descriptors are stored, allowing vendors to use those chips with their own manufacture, serial number and device descriptor strings, as well as their own device-specific VID/PID. Heck, you can get blocks of PIDs from FTDI so that you don't have to buy your own VID.
I don't know what sort of functionality does the driver use to discriminate between legit chips and copies, but it's possible that it could do something like attempt to write an EEPROM byte at an address that's too large. Perhaps on the genuine chip, such write is ignored, but on the counterfeit chip the write wraps around. That'd be an implementation bug in the chip, pure and simple. The negative effect (zeroing out of the PID) is a bug, even if it's exploited by the driver. I wouldn't shed any tears for the people who use the fake stuff. You can buy FTDI-branded serial converters from mainstream vendors, there's no need to buy Chinese copycat crap.