I want to play this on my TV with controller, but ELEX does not recognize any inputs from my controller (Sony DS4).
If I press the PS button Steam is opening, so I hope not that this is a hint how I should have bought it.

Does anybody have a idea how to get it working? For example with No Mans Sky all is running flawless.

I will test it tonight as I prefer using a PS4 Controller instead of an XBox one. Based on my very short research on the Steam forums, that version doesn't seem to support the DS4 natively.
So I guess you need to use the "DS4Windows"-tool like in 90% of Windows games.

By the way: Controller support has nothing to do with where you bought the game. ;-)

Get DS4Windows. You have to deal with the Xbox icons, but it works great this way.

Thanks for your test and link.
Nevertheless only supporting the xbox controller isn't up to date. At least the DS4 and the Steam controller should be supported.