Empathy: the psychological identification with or vicarious experiencing of the feelings, thoughts, or attitudes of another.
False empathy is claiming empathy with people one has never met/has no connection with, for the purpose of harming/denying empathy to someone right in front of oneself, someone with whom you ought to have some sort of connection.
It is generally expressed as, “yes, I see you are hurting terribly, but I do not care because of your opinion on X”, or even, “No, you cannot actually be hurting, because of your opinion on X.”
This is not empathy. It’s a form of bullying; it drives people apart and decreases trust.
(Another form of false empathy is identifying with the suffering of others, but being completely clueless about their hopes, joys, angers, humor, or basically the vast majority of emotions and motivations they feel.)