What's to stop someone from sharing their UPS Prime membership with everyone they know?
Verified address(es) sound like the only answer, and that brings up another set of problems to solve.
Beyond that, just because UPS offers a membership doesn't mean that retailers will have it integrated into their checkout flow/logic. In a lot of cases that would be a re-work and could take a long time before it was practically usable.
Maybe it's strictly associated with the address: every time UPS ships something to your address, it gets flagged for express treatment. Seller doesn't have to know about it at all.
This would actually be significantly harder than you would think - definitely much more difficult than a 'flag' on your address.
Below constitutes my (limited) experience working as an engineer at a company with a big logistics component (not Amazon).
When you run an operation the size of Amazon (or even muuuuch smaller) and are putting out a lot of freight for UPS/FedEx, you batch up your outgoing material based on ship method and 'next hop'. In order to speed up the delivery from a 3-5 day window to a 2 day window, you would have to explicitly purchase the '2 day' shipping option for that customer, or it would get batched in with all of the other 3-5 day goods, which could get sent to a totally different distribution center than the 2 day.
We're talking huge numbers trucks all going to "Indianapolis 2 day" versus "Lexington Ground" or whatever - once it gets on that ground shipping truck, I don't think it matters a ton what your address is flagged, as it's going to go that shipping method.
As andrewflnr said, it would be tied to your name and address, or maybe two addresses (home and vacation/business) and four names (family) the same way USPS offers Informed Delivery[0] based on name and address. It offers the carriers a double dipping (charge shipper standard rate and then charge you subscription fee) but there's definitely a market for it.
