Well, if you look at #13, it seems they are requesting your registration#. Please, correct me if I'm wrong.
When initially responding to a mayday in Sectors Delaware Bay and Baltimore, registration numbers are not requested by the CG over the radio. Once they arrive on scene, depending on the situation, that may be a different story and they probaby will ask to see your registration card.
It seems odd to me that the CG would waste precious time on this issue in a Mayday, if it didn't provide pertinent information. Which brings us back to the question in #5 you didn't answer. Do you know for a fact that CG doesn't have access to the registration data base (in your state anyway)? If so, how?
I am a CG Aux coxswain. The last time this came up on one of my patrols, the local marine police for the state was contacted. CG has access, of course, to information on documented boats, but each state maintains its own database for state registered boats.
Additionally, the reason that it sounds like they may be reading off a checklist is because here at least, much of the time the newest recruits are on the radio and they just may be following their checklists until they become more experienced and obtain a smoother delivery. Sector Delaware Bay is always looking for watchstanding volunteers from the Aux as resources are limited these days. A small percentage of CG members are out on the water. In Sector Baltimore there may be one CG boat covering the upper Chesapeake unless additional resources are called in for an emergency. Most of the rest don't know what the local area looks like outside of the building where they are stationed.
Providing as much information as possible during the initial call is important in case the distressed boat loses radio contact and that information is all the CG has to go on to respond. If you know your registration number great, provide it. But it isn't assumed that you'll remember it in an emergency situation before remembering your boat name. As someone mentioned, DSC connected to a GPS will eliminate most of this issue.