The connector on the PCB is called a ZIF (zero insertion force) connector. Normally they are specified by the number of pins, the pitch of the pins, and whether there is any locking feature or "ear" on the sides of the ribbon cable. It looks like a standard latching connector made by any number of companies.

The ribbon cable looks like it is custom designed for the display's electrical pin out and the mechanical design of the enclosure.

If you can figure out the mfg of the display itself, you should be able to figure out the ribbon cable pinout.

I know a fair bit about connectors and circuit fab, but not an EE so hopefully this helps!

We need a picture of the entire pcb. The ribbon cable (technically a flex pcb, not a ribbon cable) is totally custom and not something you can put a signal on. The little green pcb looks like the driver, so we would need pics of that and maybe figure out where you can input some kind of signal.

These kinds of displays can be really hard te re-use. Often their driver is just a simple blob (chip on board) which would be completely custom. This can be made cheaply to order with as few as 500 units. So mass produced readers often use completely custom stuff to get down to a really low price. Unfortunately this means it's not possible te repurpose the screens later.

Spi is the norm for hobby displays because they are made to be driven easily. There is a driver chip that receives the spi signal and drives the display. With an actual product this often isn't the case and the screen gets driven directly.

You could try using the e-ink buddy from Adafruit. I bought one for driving a price tag e-ink as well but have not tested it yet unfortunately. On their site they say that e-ink connectors are pretty standardized.

Hey, I am not 100% sure but from other screens i have seen it is probably to boost power. One of the open source ereaders has a little chip that is similure. It is probably a spi display. Do you have the model number of the panel?