The Steam Controller is actually fully integrated into the Steam Client. It has an additional USB adapter that establishes the wireless connection.
I don't suppose there is a driver for it on the PS4.
Why use it on the PS4? It already has very good controllers?
A bit pointless in my opinion.

Greetings
Peter

Not yet, but Sony has announced mouse and keyboard support for the PS4 as far as I know. With a bit of luck, you will be able to use the controller in the future. But wait and see how that develops. Technically, you would have to install a driver on the Playstation or the Steamlink would be compatible with the PS4 controller format (both of which are not yet possible / not the case).