I think that there are some significant points in the work to suggest that appearances and identity are strongly linked.  Oates is exemplary in suggesting that these constructs of identity are duplicitous, something that Arnold manipulates and to which Connie falls victim.  For Connie, the opening lines help to indicate how looks help to formulate her identity.  Her mother fusses at her for constantly looking at mirrors and compares her to her sister.  This is something that Connie understands as part of her identity for two reasons.  The first is that it reflects how much of a "square" her mother is perceived to be in that she fails to understand the importance of "good looks" and that Connie feels that since her sister is "uncool," her supposed preoccupation with her looks helps to enhance her own identity.  When Connie goes out with her friends, she exudes a sense of confidence in her own looks that helps her feel that she is in control of her social situation, the world around her.  To a certain extent, it works, as Arnold is initially attracted to this.  All of what Arnold constructs in terms of physical looks is geared towards a duplicitous construction of identity.  His tanned appearance, something that Connie recognizes is not real, the boots that cause him to wobble, and the makeup that is caking up, reflecting his older age, are all examples of this.   For Arnold, the construction of his appearance as a younger person who can ensnare someone like Connie is critical to his identity as someone who seeks control.  Connie's recognition that her emphasis on looks and the construction of her identity brought her this fate, her own doom, is one of the critical aspects in her development in the short story.  At the moment when she "no longer recognizes" what is in front of her, it might also be a moment when she fails to recognize her new physical sense of self, as what was in the past proved to be a fated construction of identity.