How to contact Apple Watch's owner to remove the Apple ID from it?

Activation Lock is designed to prevent anyone else from using an Apple Watch if it's ever lost or stolen.

About Activation Lock on your Apple Watch - Apple Support

Activation Lock is an anti-theft feature that cannot be bypassed. Only the previous owner can remove the lock by using their Apple ID & password. Apple will not help you find the previous owner. If the previous owner cannot or will not remove the lock, then the watch is useless to you or anyone else. Try to get a refund (through eBay) or dispose of the watch properly. In the future, see the 'Before you buy a watch from a third-party' section in:

What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support

If Goodwill does not have contact with the owner, and there is no information on the Watch screen from the previous owner, there is no way to get in touch with them. I would try to return it for a refund, as the device will remain useless to you, or you can recycle it.

See the 'It goes on to help the planet.' section in:

Apple Trade In - Apple

Not that can be shared, and as users here, we do not have access to that. Apple will not get involved in identifying original owners of used devices.

This is a user-to-user technical support site operated by Apple. None of us has access to that information. Either return it (Goodwill should not have sold it) or recycle it.

You can do so at an Apple Store or Best Buy.