CUC Administration >> User >> Select User >> Edit >> Greetings >> Standard. By default, callers hear option is set to System Default Greeting.
System Default Greeting—Select to use the prerecorded system default greeting. Unity Connection plays a prerecorded greeting along with the recorded name of the user (for example, “Sorry, <user name> is not available”). If the user does not have a recorded name, Unity Connection plays the user extension instead. When a greeting is enabled but not recorded, Unity Connection plays a prerecorded system greeting.
Recorded Name is what is set up as Display Name on the main user page. If you will remove the entry from there, then it will play the extension. Else, you can choose to play My Personal Recording and the user can record their own personal recording.