Can I press charges or prevent a neighbor from seeing my grandmother?

To press charges or to get a restraining order you'd likely have to prove abuse. The idea she uses a false name is troubling, however. Could you talk to the care home people and see if this woman can be barred from visiting by asking them to post a picture of her? You can explain she uses different names. A restraining order would be next, but this could get involved. I guess it all depends on how much these visits upset your grandmother and how much you are willing to go through legally.
Take care,
Carol