After a lot of time playing with this, and trying the tips larry1 suggested (with no luck for me), I think I at least figure out what is going on, and what criteria airdrop is using when there are multiple emails associated with you Apple ID.
Alphabetical.
Like many others on this thread, I have multiple email addresses associated with my Apple ID, including two for my daughters. At 8 years old, I DO NOT wish to create separate IDs for them, nor should I based on Apple's ability to allow you to set up multiple emails, and choosing which are used for Messages and Facetime. Sure, I could, but it should not be necessary, if they had just used existing ideas used for Messages/Facetime.
AirDrop needs to have a similar checklist in the settings, so you can choose which Apple ID email that device is associated with.
Instead, it has a hard-coded, built-in priority order that doesn't appear to be changeable. Instead of priority going to the "primary" email address or even the email addresses associated with your "me" contact, it takes all of the emails associated with your Apple ID, and prioritizes them based on alphabetical order.
Using my case as an example:
On my Apple ID, I have the following emails:
-- scott@ (primary contact email, the email of my "me" contact card, AND contact for me on my wife's iPhone)
-- emily@ (my daughter, used on her ipod touch for iMessage and Facetime)
-- ipad@ (for my ipad for iMessage and Facetime)
In all examples, I am requesting an airdrop tranfer from either my wife's iPhone or my iPad. My iPhone is to be the receiving device, which should be showing as "Scott"
I kept having issues with Airdrop always showing me as Emily when trying to send from my wife's iPhone and from my iPad. Both sending devices have a contact card for Emily, and a card for Scott, with no mixing of the email addresses. This is because of those three emails listed, it is the first one alphabetically that match a contact card of the PERSON REQUESTING the tranfer.
When in "contacts only" mode at least, my phone appears to be broadcasting all of the available emails associated with my Apple ID, and trying to match the requesting device's contacts, going in alphabetical order.
If the sending device only has a contact with scott@, it shows up (as I want) as Scott
If I add a contact with the ipad@ email, then airdrop shows my iphone as that contact
If I add a contact with the emily@ email, then airdrop shows my iphone as that contact
I can repeat this with the exact results each time. It makes sense, from a very simplistic coding approach without much thought given to how Apple allows us to use our Apple IDs. Using an alphabetical priority is completely non-intutive for the user, and does not allow for any reasonable customization.