Ok the short of it is, as far as consoles go at least, there's no such thing as a universal adapter..... because Microsoft (and sometimes Nintendo and now that I think of it, Sony too).
Let me explain. Starting on the pc end of things, almost all gamepads, joysticks or whatever usb devices that work on a pc are HID complaint. HID=Human Interface Device... even your mouse and keyboard, assuming they are usb 2.0 or higher, are HID compliant. HID devices generally don't need a driver on the pc to work and/or the needed driver is already included with windows. Note I said HID, not HIOD. In theory HID devices allow for some type of output standard, but I've never seen it used. In other works if you want force feedback or rumble to work... yup it needs a driver downloaded anyway. Direct Input is just an older usb joystick as all usb gamepads/joysticks that aren't xbox 360/xbox one clones are using direct input. Except for specialty controllers (flight sticks and wheels) this standard is starting to go away. The reason they still use it is the new standard (xinput) doesn't allow for the kind of outputs and extra axis these devices require. Xinput is the new standard on the pc... if you play a newer pc game it's almost definitely going to support xinput, and it may or may not support dinput. Most cheap-o Chinese usb gamepads show up as an xinput device by cloning the ID of an xbox 360 gamepad... tricking windows. Note that dinput devices and xinput devices are probably also HID devices. But that's on the pc end of things.
On consoles things are different. Microsoft has an encryption chip on all of their gamepads. Only officially licensed gamepads will work, so those cheap ones from ebay… yeah they show up as a 360 pad in windows, but not on the game consoles. Any "whatever" to xbone/xb360 adapter needs to have an official pad plugged into it to act as a pass-through. Nintendo likes to do it's own thing. Their controllers are seldom usb compliant, much less anything else. It's usually possible to spoof a Nintendo controller, but because their protocols are odd, in addition to having weird controllers, it often doesn't matter anyway. I'm not as well versed with sony stuff, but I believe most everything from the ps3 onward can be made to work on the pc and other devices via custom drivers and/or an adapter. They aren't HID compliant though.
So to summarize... almost all usb gamepads are hid compliant. That isn't necessarily useful though on the pc or on consoles. Only Microsoft stuff and it's copycats are Xinput compliant BUT it's pretty much the standard on the pc at this point for gamepads. Anything else is direct input.... unless it's by Nintendo or Sony.... Nintendo does it's own thing and Sony hates to comply to any standard that Microsoft uses (although most of their devices can be made to work anyway). So yeah, no such thing as a standard across all things.