The Status Report "Name" as well as the "Attachment" contain a date, but these do not match the publishing date.
Clarity 16.4.1
The Status Report "Name" is set once (during its creation, normally when accessing Status if there is no Draft Status Report) and can be manually edited when not published:
Name will not be updated/changed automatically (even if it has not been manually edited from its default value).
For the Status Report "Attachment", there is a setting in Classic Administration > General Settings > System Options > New User Experience: Set Attachment name to Status Report Date and Current Time on Publish.