Hi!
You're right, because you are using the attribute to get the date it works a little differently. The date formatting tags only apply to date-specific dynamic text elements like Date Exported or Current Time.
For attributes, the date format is read from the table itself. So you can edit the date format in the fields view and that will be reflected in your layout. The attribute dynamic text just displays what's in the table, it stays in sync with how the table is formatted.
If you don't want to change the formatting for the attribute table you can use Value dynamic text instead. Value dynamic text also reads values from an attribute table, but it can be used without a map series and applied to stand-alone tables. Value dynamic text supports Arcade expressions so you can write an arcade expression to display your date data differently on the page than it is shown in the table.
Cheers,
Aubri