Playstation Controller compatibility settings.
Honestly I dont exactly know what issue PS4 controllers have on PC, but they rarely work. I think its a mix between Sony not providing good drivers initially and then a lot of different fixes from other software companies because of these bad drivers.
Steam is most likely the culprit this time, so we will do some simple changes to steam to make the controllers work.
First of all, top left corner of steam, open steam settings.
Second, inside settings, navigate to controllers and make sure the playstation controller doesnt use the steam input for playstation controllers.
Important: Restart Steam After Closing Settings Window!!!
That should be all. Invector should now correctly find the PS4/5 controller.
Why does this work? The way I interpret it is that steam input takes the PS controller and says: this is an xbox controller, which obviously defeats the whole purpose. This MIGHT have been needed before sony provided correct drivers, but havent been needed since PS3 days afaik.
Caveats and Limitations
There are a lot of playstation controllers out there, if you have a special/limited one it most likely wont work. The game looks for a specific hardware ID to map PlayStation controls to.
If you follow the guide and Restart Steam / PC correctly, but still have issues, then you might have to learn the xbox icons unfortunately.