When I uninstall a Device, I always check the box that asks if you want to uninstall all drivers with it. Maybe that would keep it from coming back, but do the mice have special drivers or software loaded on the system? Something might be reloading the drivers....
I haven't bumped into a situation such as yours. If I just remove a mouse, it normally goes away.
Those Unifying Wireless Mouse/Keyboard receivers normally have software that can adjust the configuration. Perhaps open that panel for Logitec and see if the Unifying option would allow you to unlink any mice that might be associated with it and the keyboard. I don't have such a system on this computer, but I can check it on another one.
If you have a wired mouse keyboard, you might even unplug the Wireless receiver and see what happens. It may even need to be uninstalled to correct possible record keeping problems.