But for the types of jobs I have had, oil and gas exploration where there are corporate secrets and proprietary information worth millions and sometimes billions of dollars, federal government service where there can be national security issues, and corporate banking, which obviously involves many fiduciary issues and where honesty and integrity are a must, I don't think it is inappropriate at all.
Everyone having access to the private government information and/or information worth billion of dollars must sign papers about not discussing or using it at least every year (and sometime even more often). Moreover such companies usually keep really good track of who's doing what on their computer and when/how reaches the database.
With that being said do you really believe someone can't create a 2nd account (email, FB, you name it) company doesn't know about and just use it? Of course people do. I have 6 email accounts, for example: 1 for friends, 1 for orders, 1 for Craigslist and ads, 1 for horse and pet-related stuff, etc.
But I still find it outrageous if I come to the interview and will be told something like "you have to give us all passwords now, so we can check your correspondence before we decide to hire you". Or (even better) send them all even before they consider me for the interview.
BTW, I went through background check in past, and it's a really intense one when they talk to neighbors, co-workers, friends, etc. (besides checking records, of course) - in many instances something would of come up if you would be unreliable or potentially dangerous.