The guide for using a PS4 controller on a PS5 does not work if the PS4 controller is currently registered to a PS4 device. I did not see the way to handle that in the instructions, but it is simple:
1. Use a PS5 controller on the PS5 to go into Settings, Accessories, Bluetooth Devices.
2. Hold down Share and the PS button at the same time on the PS4 controller until it blinks white.
3. The Controller will appear on the Bluetooth list on the PS5, you can select it with the PS5 controller and pair it.
If you don't follow these instructions, the PS5 will claim it cannot work with the PS4 controller, but will not explain why.