As a follow up to Rickair7777, I think all his statements are correct. The example "Prime Air" is what the public see's as an airline. My understanding is that "Prime Air" is a corporate name and marketing tool. There is no Air Carrier, (yet), called Pirme Air. There is Atlas, and Southern Airways and other carrier's who are certificate holders and do business with Amazon, and operate as "Prime Air." I don't know if this is viewed by the FAA as a "code share." I think "code share" only applies to an air carrier who holds themselves out to the public for air transportation.
You cannot ship via "Prime Air" unless you have purchased something via Amazon. (At least I don't think you can.) Prime Air operators have a private agreement with Amazon to carry Amazon freight but Amazon does not have an operating certificate.