As a former WDW manager (among other things!) most Disney employees are required to wear their own name tag with their real name or nickname on them. If a CM loses their name tag, they can sign one out from a manager or a coordinator until they receive a new one. We used to track who had what name tag. And yes, if they continue to not have their actual name tag, it could result in disciplinary action.
@Jennasis is correct, the only place CM’s may not use their own name is at the call center. No two CM’s there can have the same name, this is for accountability, that way if a guest says they spoke to “Charlie” , there is only one Charlie, easy to identify. If their real name is not taken and they want to use it, they can. Otherwise, they are shown a list of names and they can choose. Regular park or resort CM’s use their real names or genuine nicknames. They can’t just make something up and change every day.