She did not want to be known only as the wife of her husband. She insisted on having her own 'identity'. [=on being known for her own qualities, achievements, etc.]
The face of the witness was hidden in order to protect her 'identity'. [=to keep her name from being known]
He has a strong sense of personal 'identity'. [=a strong feeling about exactly the kind of person he is]
(..) at the supermarket? Come on. Who'd want to go shopping as Elastigirl, know what I mean? Superladies, they're always trying to tell you their secret identity. Think it'll strengthen the relationship or something. I said, "Girl, I don't want to know about your mild-mannered alter ego." (..)2004 The Incredibles
(..) I can break through walls, I just can't... - That's fine. - I can't get this on. Mr. Incredible... Do you have a secret identity? Every superhero has a secret identity. I don't know a single one who doesn't. Who wants the pressure of being super all the time? Of course I have a secret identity. (..)2004 The Incredibles