Yes, the plate number on your vehicle is publicly available data. But I don't see how that is very usefully. I'm not sure there is any way for a random person to do anything with a plate number. It would require the person to have some sort of connection to the DMV or other agency that has access to your name and address based on the plate number. And those people are still not going to have access to your medical records even if they did manage to get your name and address just by knowing your plate number.

EDIT: the bit about getting at medical records seeped in from reading another comment.