Right, and to keep this analogy consistent, do storage owners routinely open all storage units, then open up all your boxes and search through them to check for flammables?
No, but that's primarily due to a lack of interest/manpower. They (probably) reserve the right to ensure that you aren't doing things that are against their TOS. There's (probably) no right to privacy.
They don't give out free storage. If you aren't paying with money you're paying with something else and you have no expectation of privacy when you agree to terms that say as such.