Actually, that's why it DOESN'T make sense. It's something that was used to demean and dehumanize her, and reduce her to being nothing more than a thing without agency. It's something that was used by the Facility to torture, abuse, and torment her, and deprive her of her sense of identity and self worth. It'd be like if Magneto started calling himself by the serial number the Nazis tattooed on his arm.
It works for marketing, yes, but not as something she would continue to use in-universe.