The PS5’s DualSense controller doesn’t appear to work natively on the PS4, but interestingly it connects to the PS3. When PS5 peripherals went on sale in the US, YouTuber Midnight Man managed to test the DualSense. In the video, you can see that the PS4 does not recognize DualSense using a USB connection and it recognizes a wireless controller in the Bluetooth menu but does not allow control once connected.
However, when connected to a PS3 using a USB controller, the DualSense is recognized and can be navigated through the console menus. ResetEra user iceblade previously said that when connected to the PS3 this way, not only the PS button works, but it can also connect wirelessly.
There is a workaround to use DualSense on PS4. If you connect it to a PC or Mac via USB and then use the PS4’s Remote Play function, the controller will work as expected.