Is there a service to accompany my 16-year-old daughter through Athens airport that's not the airline?

I’m afraid it seems the airport relies on airlines providing this service (see https://www.aia.gr/traveler/travellers-info/special-assistance).

As some airlines do not provide an escort service for UMs, you are faced with two options:

• book with an airline which does provide the service
• or give all the useful info to your daughter so she understands it’s really not that hard.

Explaining to her what she needs to do, showing her the signs she must follow, and letting her watch videos on the topic should help her a lot. It's really not that hard:

• follow the herd out of the plane until inside the terminal (this may just involve walking up a jetway, or that could involve walking down the stairs, taking a bus, and walking a bit more, but everybody else will be doing exactly the same thing, definitely nothing to think or worry about)
• follow the signs that say "Baggage / Exit"
• At passport control, enter the queue for "EEA visitors" (the language may be "EU visitors" or "EU passports" or "EEA passports" or something similar). It doesn't matter if she enters the other queue, it will just take a bit longer.
• when she gets close to the counters, wait for a counter to be available to approach
• show her passport to the officer and answer any questions (there probably won't be any)
• look up on what carrousel the luggage is to be delivered. Remember to check using the airline/flight number and not just the origin airport, as there may be multiple flights from the same airport arriving at more or less the same time (and airport names are shown in greek half of the time, which makes it a lot less practical)
• wait for luggage to come (can be a bit long sometimes)
• pick up her luggage and double-check it's hers
• follow signs for "Exit"
• go through the Blue channel (EU arrivals)
• look for whoever is waiting for her!

It seems like a long list, but I really went into a lot of detail. The short version is really: go to passport control, pick up luggage, exit.

Also, she has the advantage that she speaks English, so in case she has any issue she can just ask anyone for assistance.

I recommend you make sure she has international roaming activated on her phone before leaving (this may require an option to be activated on her contract and/or activation on the phone itself, sometimes separately for voice and data). That way she will be able to call you or even use video calling to show what’s around her for help.

Another option is when dropping her bags at the origin airport, you may try to ask someone else on the same flight (and exiting at Athens as well) if she can tag along. Families with children are your best bet, as travellers going alone may not want the pressure and/or delay. In some departure airports it might be difficult as there could be many flights to very different destination dropping luggage at the same counters, but it never hurts to try.

She could also ask someone on board (like the person sitting next to her). If she feels too intimidated to do it herself, just have her call you once she's on the plane and ask her to give the phone to her neighbour and sort it out for her. Don't forget to check the person is indeed exiting at Athens and not connecting to another flight.