I think it varies between places, but I don't think it's too uncommon for them to ask for proof of insurance and registration to be kept on file. Not that I am for giving away info that isn't needed, but I seem to recall a while back reading about how some folks were using storage places as meth lab kitchens. Could be just a precaution for them.
In the end, if the storage place was good and I felt it was the best place to store, I'd have little reservation. If they asked for SSN, DOB, specific work info, salary, and other specific items, I would not provide those.
Computers manufactured by companies such as IBM, Compaq and millions of others are by far the most popular with about 70 million machines in use worldwide. Macintosh fans note that cockroaches are far more numerous than humans and that numbers alone do not denote a higher life form. -NY Times 11/91