micky wrote:

The devices that plug into the cigarette lighter and play sound from a
USB drive through car radio speakers have the ability to remember where
they left off,

if they are programmed specifically to do so.

and, even when power is removed from them when the car is
turned off, they restart from the start of the file or segment, often
the start of a song, that had been playing when the car was turned off.

if they are programmed specifically to do so.

My question is, How does the device keep track of where to start up
again.

they are programmed specifically to do so.

Is there a file written to the device that plugs into the
cigarette lighter, or is it written to the USB drive?

There is hardware allowing both to be programmed for that purpose

If to the device itself, will it look for a specific file name if a different USB drive
is plugged in, or can it be fooled then into trying to play song 6, for
example, on a different usb drive.

Computer viruses can sometimes re-program a USB or other device to suit its purposes. For example, notice that font types, sounds or colors may change seemingly for no reason.