I imagine that at the time of the accident name and address will suffice, and indeed many drivers do not have their insurance details at hand. Subsequently, if insurance details are requested, they must (presumably) be provided.
Incidentally, my own claim, which was the subject of the original post, seems to be progressing without any action on my part, as I have received a letter from a firm of Automobile and Claims Assessors instructed by the Hire Company's insurers, with a view to inspecting my vehicle.