In order to sue the City, you need to be able to show what damages you incurred. Additionally, there is a level of immunity given to government entities that typically protects them in situations like this. You can make a complaint about the incident and present the information to show you are innocent of the violation. If you would like more specific advice about your circumstances and the likelihood of prevailing in suing the City, you should consult a local attorney.
The information provided does not constitute legal advice and is presented without any representation or warranty whatsoever, including as to the accuracy or completeness of the information. Always contact an attorney when seeking legal advice.
I know it's frustrating, and you will probably have to go to court and show the picture to the prosecutor. If it's clearly not your car then they should dismiss the charge. It will be an inconvenience, but I don't think that the City would waive it's governmental immunity over such a suit. Like you said, mistakes happen and the level of damage should be a couple of hours of your time. There should be an easy enough fix for it though.