To my understanding, the remote is programmed to the computer in the car, not the chip in the remote. correct?

So, if that is true. if you have a remote with the same FCC_ID and IC number, the remote should be programmable to the computer in the car (i.e. Camry, Prius, ...etc as long as it is a smart key)

the KEY words being having the correct FCC_ID and IC number.