Apple 8watch
How to register applecare purchased from costco
Apple Watch Series 8, watchOS 9
Question marked as
You should not need to register an AppleCare+ plan you purchased as at purchase it is tied to the serial number of the device. It may take a couple days for it to show up on the watch, but again it isn't something you need to register, nor can you register it.
If after a couple days you don't see it by going to the Watch App on your iPhone, General About - Under your name, you should see AppleCare+