iOS Chrome has "Incognito Mode*". The asterisk is there because WebKit does not have any way to make localStorage private, meaning it's shared between Incognito and Normal browsing modes, so there's a possible leak of information there, especially if they're using a third party tracking service. This is true for any of the private web surfing apps that use WebKit.
It only clears your browsing traces locally (history, cookies, cache, etc). Everything else can still be tracked on server-side. BTW, every private browsing mode that I know works pretty much that way.
In the current model real privacy can only be achieved running something like Tor.
