I think older versions of InDesign allowed customisation of the .mrk (mark) files that controlled what got spat out when you ticked Page Information in the Marks and Bleeds PDF export.
As far as I know, Adobe has moved away from editable mark files. Now, the Page Information data is hard-coded into InDesign, and users have zero official control over the content or formatting. It will always output the document name, page number, date, and time, in a rigid style.
What you can do
Create your own custom page info by adding the Slug Info in the Document setup and then include that in the PDF export
You can create a parent page item in the slug area.
Use variables like File Name and Modification Date (customise as you see fit)
Combine them with custom text like Confidential or client names.
You might be able to even script it if you want a dynamic solution that updates automatically on save or export etc.
InDesign Variables
Go to Type > Text Variables > Insert Variable
Use File Name, Last Modified Date or create your own custom variables.
Acrobat stamp
If you're in a pinch, you can also post-process the PDF in Acrobat using a custom stamp, but that's more of a hack.