At our hospital, the security picture name tags have only first names and the full name is in small letters on the back. It's definitely in the name of security and confidentiality.
On the other hand, each individual is at her/his own discretion to provide her/his full last name and I would have no problem doing so (though can't speak for anyone else of course). I have honestly never noticed whether or not the doctors have just their first name on their picture ID but I know the other hospital professionals (specifically, OT, PT, ST, Dieticians) have first name only. I will try to look and see next time I run across one.
Actually, speaking of which, our hospital REALLY needs to step up and ensure people are wearing their name tags in a visible place -- many people stick it in their pocket or wear it flipped over. That is the part I don't understand!!