I'm sorry, I forgot to mention that you have to press the PS button once after plugging it in. Did you do that?

If that doesn't work try restarting Steam and look into your controller settings if it works there, if it does you're good to go

Keep in mind that many games don't natively support Playstation controllers. You can work arround this by emulating an Xbox controller instead (enabling your PS4 controller in Steam should do just that for Steam games). Or alternatively do this regedit edit in your games Wineprefix to achieve the same in any launcher of your choice

If all else fails it might be worth trying out ds4drv

Good luck my guy and (hopefully) have fun gaming! :)