I have the exact same problem. I updated my two iPads and iPhone to 10.2 and the iMac and 2013 MacBook Pro to 10.12.2 and now cannot Airdrop any files to the three iOS devices. No problem Airdropping to 2013 MacBook Pro or the mac mini. All three iOS devices Airdrop to the Macs as designed. My 2016 MacBook Pro (arrived yesterday) is still on 10.12.1 and does Airdrop as designed.
In short, it is the older MacBook and the iMac (27" late 2013), both on 10.12.2, that won't behave properly. Yes, all the usual things have been checked with regard to who is allowed and Bluetooth being on, etc... Could there be a change in Bluetooth brought in by the 10.12.2 update that is upsetting this particular Apple cart??