If the pet never had a microchip I would think that would be possible. I am not certain tho- if the dog is already chipped, if the information can be changed or transferred without the registered microchip owner's consent. And you cannot put in a new chip, because the chip reader would blip trying to read two of them.
You raise and interesting question tho, I know breeders should certainly register the chip on behalf of the owner with their name as a backup, and NOT leave it up to the owner.
So this would also be a great suggestion for rescues to do too. Even with all the best intentions and home checks rescues go through to rehome dogs, unfortunately they will always be going to humans, who you can never 100% guarantee they will forever follow up with 'their' end of the care of the pet in question.