I agree with this. We already have to agree to open up our personal and financial histories to be vetted and screened as part of the hiring process. There shouldn't be anything available on the internet you wouldn't want people to know about.
First, it's bad security to hand over your passwords to anyone without a court order. What's to say the city doesn't accidently post them to the web, as countless govt entities have done with personal information and social security numbers? Or that someone working for the city won't abuse these passwords by posting under an applicant's account?
Anyone can READ public information on the internet. Handing over read/write abilities to any govt entity on a voluntary basis is stupidity that is second only to Darwin Awards.
IF they wanted to, they could ask for the location of any interaction on the internet and your particular username. I find such requests distasteful and wouldn't personally do so, but to each their own. Mind you, I don't have overly high standards, I went through a TS (SSBI, not SCI or lifestyle poly) clearance background check. I'd never voluntary give any govt entity the ability to read my mail, electronic or paper. Any information I post publically is sterilized to my comfort level for anyone to read. What anyone else chooses to put in the public light is their business.