Please fill out your timeline: https://www.visajourney.com/timeline/profile.php?id=490235
You should probably go back and look at what the entire immigration process looks like. The instruction you've cited above applies to individuals who are citizens, permanent residents with a valid greencard an/or extension letter or temporary workers with valid work authorization. Your wife is none of these.
A K-1 visa is technically a non-immigrant visa. Your then fiancé was inspected and paroled into the US for 90 days. She had 90 days to marry you and file to adjust status. Once the I-94 ran out, she entered a period of authorized stay. That's it. Nothing more. She currently has no immigration status.
A SSN is government issued, unique identification number issued to US citizens, permanent residents and ELIGIBLE temporary workers. Your wife is not a citizen, she is not a permanent resident (yet) and she is not a temporary worker. She is a visitor who was authorized to enter the USA.
The only reason she was issued the SSN in the first place and the reason it states the restriction of "VALID FOR WITH WITH DHS AUTHORIZATION ONLY" on her card is because the assumption is her status will be verified - that the marriage occurred within the 90 days, that the AOS was filed and that the I-765 was approved. So technically, her SSN isn't even valid at the moment. It won't be valid until her case is adjudicated or work authorization is issued. This is also why it is recommended to file the I-765, irrespective of whether or not the immigrant plans to work.
If you look at what is REQUIRED for a SSN, the SSA clearly states they will verify immigration status. The only way a SSN will be given outside of immigrant status is if the applicant has work authorization. Your wife does not have any immigration status at the moment. She has an active application to adjust status. It hasn't yet been adjudicated. You didn't state whether you even filed an I-765, but based on the comment "she is NOT trying to work" and no mention of work authorization, it is assumed not.
Please read ALL of the following after selecting adult, corrected, non-citizen on the left side of the page, paying close attention to the blue note box toward the bottom: https://www.ssa.gov/ssnumber/ss5doc.htm
Noncitizens: https://www.ssa.gov/pubs/EN-05-10096.pdf
Until all conditions are met, the SSA will not update her card.