I don't know about the mainline CoC, but the ICC do believe they are God's sole remnant on earth, as evidenced by the fact that there is only one congregation in a given geographical area. For example, there is "the Tampa Church of Christ," "the South Florida Church of Christ," "the Orlando Church of Christ", and that is the name they go by.
I know I belonged to both. And BTW, the ICofC do have different names rather than by city. Both the mainline CofC and the ICofC believe they are God's sole remnant on earth.