RBBM
I don't find the real Cheryl's statement odd at all.
The phrasing in the article is a little awkward but to me it reads the police found an identification card with Mercedes picture and Cheryl Ann Wick's name. When I was 18 I was able to get a "replacement driver's license" with no type of documentation at all. The license was my photo with my cousin's name and information. This was in the 80's.
As for missing the social security card, I wouldn't notice it until I had to actually present it to someone. Even now, how many times a year do you have to present your SS card And if I couldn't find it, I would have thought I misplaced it.
With a picture ID and a social security card EDJD would have "legitimate" documentation to open accounts, work, apply for governmental assistance,etc. With those two documents she could get whatever she needed.