Except if there is the possibility of it happening without their knowledge/consent, the other person could use even the name for further social engineering. It's better to not give out any information automatically. Granted the user has to approve a Name Drop share but the screen does display the user's contact info that would be shared either way, so if the phone is visible to the person trying to obtain the info, they'd still be able to see it even if the target doesn't approve the share.
It is a bit overhyped since it's not like someone shady can go around sniffing everyone's contacts automatically, but it's still worth tuning off for anyone who is privacy or security conscious.
If that is the case then it's better but I'd still shut it off and err on the side of caution. The Apple demo video does not mention needing Air Drop enabled, only that both users need to be signed into iCloud: https://www.youtube.com/watch?v=aZL5D1k-4aI
It also requires accessing your contacts database, which is encrypted on iPhones...
Because it's encrypted, it's impossible to share contact details unless someone enters the device passcode (or else does a biometric unlock - which effectively stores your passcode temporarily in a secure location that is wiped whenever the device is powered off or left unused for several hours).
Somebody who clicks "accept transfer" on the screen without knowing why it popped up deserves whatever comes next. Only exception being young kids who shouldn't have access to a fully functional device anyway. If there's some sort of "Toddler mode" on iPhone, then yeah def have airdrop disabled when in that mode. This is a parenting issue. We should be far more concerned about child advertising and parents putting their entire kid's life story on Facebook.
Also, why are police fear mongering on social media in any official capacity. Seems pretty damn unprofessional.