Why can't my wife's iphone see my phone on Airdrop when I set it to "Contacts Only"
I just solved this problem after google brought me to this question. 
This problem has been bugging me for months. 

My wife and I couldnt airdrop to each other unless we set airdrop to everyone and then we could send if we showed up in the "other people" section when using airdrop. 

In my case I we had been using one apple id across both our phones and after we setup an appleid for my wife there were some issues. 

I'm not sure if all of this is related but it seems that AirDrop is a feature of or at least linked to contacts. So after checking that all of our iCloud details were distinct I started to look at contacts. 

Not sure if this is directly related but her 'My Info' in Settings/Contacts was still set to me. So I fixed that. 

Did a restart of both hers and my phones. 

Then she had multiple contacts for me. I beleive this is due to her and I working at the same organisation and both our phones being managed by the corporate MDM. When trying to link the two different contacts it would show a third contact with one of her email addresses, like the second contact had some link to a third contact that I couldnt see in her contacts list. Seemed a bit like an errant record. So in the end, I deleted all  contacts entries for me from her phone and added myself back using the minimum appleid  email address and mobile phone number. 

I also deleted the instant messaging details for each other from both of our phones contacts. 

After this I got the 5 airdrop requests that I had tried to send to her contact from my phone. 

This is a bit of a ramble and I'm not sure if all steps are required for others in this situation but figured more information for others struggling with this issue.