Why Is My Controller Detected Under A Completely Different Name?

Both halves of this have a clear explanation, and only one of them is a defect.

The name is deliberate and not a problem. Windows games overwhelmingly read controllers through XInput, which was built around the Xbox 360 pad. Any third party controller that wants to work in games without shipping a driver declares itself as an Xbox 360 Controller for Windows, because that identity guarantees XInput picks it up and every game treats it as a standard gamepad. Almost every PC-compatible third party pad on the market does this, including good ones. So your controller is being honest about the protocol it speaks while borrowing someone else's name to do it, and that is why your buttons work in games at all.

The wrong button positions are the actual flaw, and they are the pad's firmware, not your system. When a controller claims the standard gamepad mapping, it is promising that its buttons sit at the indexes the specification defines: index 0 is the bottom face button, 1 is right, 2 is left, 3 is top, and so on. Budget manufacturers sometimes wire the buttons to the wrong indexes and ship it anyway, because most games are forgiving enough that nobody notices until they use a tester. You noticed. There is no setting to fix this, because the controller is reporting what its firmware tells it to report.

The practical options are remapping software on the PC side, which handles this cleanly once configured, or returning the pad. Which is reasonable depends on how much else about it is good. The one thing I would say is that if two buttons are mapped wrong out of the factory, it is a sign of how much testing went into the product, and that tends to show up elsewhere too.