The Apple Care Protection Plan and Apple Care + are accessible for Mac, iPad, Apple TV, Apple Watch at the Apple Online Store, Apple Retail Stores, and numerous Apple-authorized re sellers and third party apple technical support providers. Apple Customer Service will have limited number of alternatives accessible in the nation or district where service is requested.
As most AppleCare these days is actually AppleCare + (covering accidental damage as well) you need to prove your device is not already damaged. This usually involves taking it to an Apple Store and purchasing it there - and they will use the serial number to register it.
I believe there are also ways of doing this over the phone for some things, like desktop computers, and again, the serial number would be registered.
There is no way to buy AppleCare in a box and attach it to a device.
If you Purchased AppleCare at the same time you purchased the device from Apple, it is automatically registered.