You can do this globally, but not for each individual. If the goal is to hide the individual's personal information, there is now an option available in the chat builder, on the display tab, that can be selected to allow company branding to override displaying individual user information. Thus, protecting employees from having their personal information exposed to the world. That said, this does cause all users to display the same generic information, which could be a tradeoff that isn't worth making depending on the situation.
This link will take you to the KB doc that confirms and outlines detailed steps. I've also included screenshots below:
I hope this helps you, or future users with the same question. Happy Chatting!