How do I register a new wireless controller to my microsoft account?

Yes, three specific controllers are all registering as the same controller. I plug in the one "good" controller that is problem free and it registers as controller 1. Then I plug in the "bad" controllers and each one will register as controller 2. However,
if I connect the wireless controller while the "bad" controllers are connected it will register as controller 3. The problem is with the three controllers rather than with the Xbox 360 itself.

As far as clearing the cache...the Xbox is brand new. I had this problem right out of the box before I had done anything with the machine. Investing in new controllers though? These things are brand new purchased from a retailer that gets them directly from
the manufacturer (Microsoft).

It is stated on the packaging that these controllers come with a 1 year warranty and the manual guarantees that they will be replaced if they suffer from a factory defect. The inability to use these controllers is a defect and I'm most certainly still under
warranty considering I just bought them less than a week ago. I'm not going to just discard these controllers, forget about the problem, and buy new ones.

The biggest question is whether or not I can do something on my end via software to fix the hardware problem; or do they need to be replaced outright?

Why do I need an adapter? I bought a xbox one wireless controller to use with my windows 10 pc. So i tried registering it and above link says Unable to register your product.

If a controller is already plugged in, no matter what it'll register that as #1. Unplug the old controller before turning the console on and then re-plug it back in and it'll register the new controller as the #1 controller.

I use a wired controller, so it's the first thing the console goes to when starting up, even when I use a wireless controller to turn it on.

(For some reason it didn't let me edit my old comment so I had to make a new one)

Here is some troubleshooting:
support.xbox.com/.../wireless-controller-functions-improperly

Important note: You must register your Xbox One console to replace a wireless controller that’s under warranty.

If you're still unable to figure it out, calling support may be a good option. It's a possibility they could just register it for you.