Catalin, consider posting in the Mac Forum as well - this issue seems Mac related as you will see below.
When using AirDrop on my MacBook Pro (El Capitan, 10.11.3) my 4 iOS devices (all on 9.2.1) do not display their given names as shown in Settings > About. They used to with previous OS X versions but they no longer do. Instead AirDrop Mac displays the names as shown in "My info" (my card) in Contacts. For instance, my iPhone does not show up as XXXX iPhone, it's given name in Settings > About. It shows up by the name under Settings > Mail, Contacts, Calendars > My info which is different. So look there on your iPhone and see how your name appears. Also go to your Contacts and see if that "weird" name appears there somewhere. If you are syncing your contacts with iCloud, they would be the same in your Mac and your phone but look in both anyway.
AirDrop is displaying differently in IOS vs. OS X. In the past they behaved the same. If I try to AirDrop from one iOS device to another, i.e., my iPhone to my iPad, the name AirDrop displays is indeed the given name in Settings > About. However, as I mentioned above, this is not the case with my Mac any longer. It appears that the latest OS X update changed this process and like I first suggested, the Mac Forum may yield other options. Good luck.