The paradigm seems very similar - I go to a service provider, pay them money to give me certain amount of private space, put my stuff there, lock it with a key and go on my merry way. When I want to get my stuff out, I go to the location, unlock, get the stuff, re-lock and go.
US laws seem to have very strong protection against someone going and taking my stuff from there. Even if the bank or the storage place go bankrupt, I'm fairly sure no one is legally entitled to go through the stuff that is being stored.
Further, if one of the bank's or storage place customers happens to store something illegal, law enforcement typically needs a warrant to seize it. However, in no way are they entitled to take or destroy property of others, not relating to the warrant.
The ONLY meaningful difference seems to be that they can't easily just access the "storage box" associated with the warrant, and servers are much more portable, so they feel entitled to just take the entire thing. I guess that if your bank had no way of opening the lockbox, law enforcement might feel entitled to take the entire vault...