USCIS no longer sends the updated card / they do update in their system but person must fill out the form
Form SS-5, Application for a Social Security Card. You will also need to provide supporting documents, such as proof of your identity, proof of your U.S. citizenship or lawful immigration status, and proof of your legal name change (like a marriage certificate or divorce decree). You can submit the completed form and documents online (if eligible), by mail, or in person at a local SSA office.
https://www.uscis.gov/tools/uscis-tools-and-resources/immigration-documents-and-how-to-correct-update-or-replace-them
Update your Social Security card
See the Social Security Administration’s Make Changes to Your Record or Replace Social Security Card webpages