I have heard through the years of quilters using the microchip method for their quilts.
The chip is the size of a grain of rice, inserted under the dog or cats skin between the shoulder blades. It does not move through the body nor is it eliminated from the body.
As to the quilts, you would choose where to get the chip, pay the $45 or or whatever they charge now, and when the person installs it in your quilt, you choose a spot, and down in the layers of batting it goes. Unless you are washing your quilt every week or two, it would stay there for the life of the quilt.
Yes you need a scanner. You can buy them from on-line pet supply stores for varying prices. Some are universal scanners, some are specific to the brand chip used. Usually only vets or shelters have them.
I personally like the idea of taking your picture with the quilt as backdrop in your home or yard and a current newspaper with date showing for verification should the need arise.
As to on on-line registry service, I guess they might need to be bonded and insured and charge a fee. After all, they will have your personal information, and you want to protect that too.
I just wish we did not have to be concerned with such things. So sad.