All of the ships that I have been on have the overlay feature on the ECDIS turned off. It tends to make the screen too crowded in busy ports. If the mates need to look at someone's name they just compare the reported position from AIS to position on the RADAR/ECDIS of the ship they are trying to contact.

Also, the AIS data is self reporting so if they forget to change the information then it's useless anyways. I've seen 'anchored' ships driving away at 10 knots.