Wow, knowing that kind of person would be very creepy.
Just like any other disorder, it's a sickness which needs help, and needs to be acknowledged and addressed. The problem is that most people only think of themselves as victims of these kinds of people, and don't take the opportunity to consider that some (not all) mimics are victims themselves. Some may've been raised in an environment where their self-concept was poorly developed, and their entire identity was defined by others. They weren't taught or allowed to develop a personal identity outside of what was expected of them, so they learned to adapt themselves, to become like those around them who seemed to be independent and successful.
Now, don't get me wrong, some of these people are just classically genius manipulators, and simply become, to use a tricky analogy, like leeches, developing a type of co-dependence on the persons they mimic. Now, if you realize what they're doing, and yes, they can be very creepy, then the best way to respond is distance yourself, without making them feel weird or rejected. They may also develop feelings for the person they're mimicing, especially if the person is nice and very caring. Now, these mimics are sometimes very sensitive and brittle, because their sense of self is shaky. So, how you respond to them is very important, especially if you want to guard against any further contact with them that could make you even more uncomfortable.
Some think they're inferior or not good enough because they don't have or demonstrate the qualities which are respected and considered positive. So, it's tricky.
If nothing else, seek a licensed counselor on how to handle it.
But that's my two cents.