Hi David,
What you see in Arccatalog or in screen capture you provided, it is fully qualified name. It shows, the feature class "NewJersey_Counties" owned by "DBO" and created in geodataabse named "Data_Library". This is by design.
Here is another old thread which may be useful:
https://community.esri.com/thread/65738
Question is why you want to strip the database name and owner name from feature class. What is the holistic requirement or what you want to achieve.
Thanks,
Biraja