Webex Calling does not have a way to block Caller-ID for the entire org. What I would recommend is both not list the names in the company directory and maybe do not use full or complete names in the directory. The other way (and I don't know if Cisco will let you) but when you set up your company in Control Hub, use "Unknown" or some other non-descript identifier so that when you do call someone it shows that instead. Being that this is the setting you want, it shouldn't matter if your company name is correct or not in the system.
