Your best bet is to just place an ad on Craigslist and in the paper in the lost and found sections. If someone tries to claim the phone, have them identify the picture on the lock screen.
I used to work at an Apple Store Genius Bar. We were not allowed to give out any information from our system ever. If you're located in the US and the phone has a SIM, it probably has AT&T as a carrier. They will not be able to give you any info on the phone either.
Just to provide some insight as far as why Apple can't help you contact the whoever might be listed in their system, remember that a computer or iPhone is a transferrable device. We would frequently have customers bring in a device for repair that was purchased secondhand and the original owner's information was still in our system.
Let's say Tommy buys a MacBook Pro and registers it. Later, Tommy sells his MacBook Pro to Kate because he's broke. Kate purchases the MacBook Pro but does not update the registration. A month later, she leaves the computer in an airport and you find it. You take it to the Apple store and they say, "Oh, here is all of Tommy's info. We'll just call up Tommy and he can come get his MacBook back."
Tommy is not a very scrupulous person, and he fails to mention that he sold the MacBook Pro a month ago. He comes and picks it up and now he has Kate's money and his old computer back. Sweet deal, right? So you can see why this isn't something that Apple would be willing to do.
The Apple employee you talked to was not trying to be obstructive when he wouldn't help you out, but there's a lot of liability things at stake here. They were pretty committed to the idea of protecting customer data, sometimes to the point of idiocy when I worked there, so they're pretty much a dead end as far as returning the phone.
What you describe is consistent with an iPhone that is still locked to AT&T as the sole carrier. You can go to the AT&T websites and fill out a form to see if the phone could be unlocked. If you can't get AT&T to unlock the phone, then you could take that up with the seller of the phone or perhaps see about jailbreaking the device and unlocking it by force as opposed to by social convention.
Best Answer
There is a cable at all Apple Stores that can obtain the serial number from the great majority of no power iPhones when connected to a specific application. Not all no power devices are able to connect, but upwards of 90% do, in my experience.
The only real issue I can foresee is that technically speaking you should only bring one device per appointment. That said, I really can't imagine anyone refusing to even check the serial number on two phones in one session, especially since you're only planning to get service on the one that is in warranty.
Oh, and be prepared for the no power iPhone to be gone for 10 days or longer undergoing screening before replacement if you ship it from an Apple Store. You can blame years of rampant warranty fraud for that.
If the technician is able to obtain the serial number of the device that has AppleCare for you, you would actually actually be better off getting service through AppleCare over the phone once you have that serial number. They can ship you the new phone while you wait for the device to get through screening at least. Unfortunately, they do put a hold on your credit card, but if the device passes through screening, you are, of course, not charged.
If you can't get to the stores, you get one shot as a consumer seeking warranty service to identify an iPhone 7 or later that won't turn on when connected to power or a computer.
The one item that's externally serialized is the SIM tray.
If you don't have confidence which tray belongs to which phone or it’s lost, Apple or an authorized tech may need to disassemble the device if they need to establish the serial in question before offering warranty service eligibility.
Older iPhone had them engraved on the case as well, but that could wear out, so all Apple warranty providers are well trained in how to help you establish eligibility for repair if need be. It might delay the repair, but you won't lose service due to this alone in my experience.
A long shot might be if you know the phone number for the phone, the carrier might have records of the IMEI / MEID / CDN / ICCID that would help Apple locate your serial. Again, this like the packing slips or packaging or receipt won’t help you if you don’t know which phone is which.