I don't think that's what's going on, for two reasons. I looked around in the options menu, and I don't see anything there to let you use any controller. And the way this seems to work is that the app itself acts as a Dualshock by logging into your profile. That's why when you try to log into it again using the Dualshock, the app disconnects from the PS4 the same way a Dualshock would if another logs into the same profile.
I'm not an engineer on their development team, but I'm pretty sure it doesn't have to work this way (that the app emulates a Dualshock) in order for the content of your game to be streamed to the app. So perhaps they'll add that option if people request it.
Can you connect anything to an iPhone? Afaik there is no usb in apple products... micro usb to lightning cable would be needed.
On windows you simply plug the ds4 into your laptop / pc with the charge cable. You can even play multiplayer split screen although the second controller connects directly to the ps4, can't have 2 connected to windows afaik. So yes, the app logs into your ps4 and you can't log another ds4 into the same profile. You can log the ds4 into a different profile and play as player 2 ...
