After struggling with Gemini for an hour, I was finally able to get a valid response (thanks to the screenshots you shared).
This is a brand-new, automated in-app promotion pushed directly by Google to Workspace for Education accounts. In fact, it is so new and unexpected that universities (like Brown University) just had to post status alerts to their IT dashboards yesterday (May 21, 2026) because their users suddenly started seeing this exact banner and calling the help desk!
Because Google recently updated the Takeout Transfer tool on May 1st to include Google Photos, it appears they have silently rolled out a global, automated pop-up to Education domains to remind users to take their data with them at the end of the school year.
It is algorithmic, not a setting: There is no explicit "On/Off" switch for this specific banner in your Admin console. Google's internal system determines when and to whom it displays these promotional banners (similar to the "Try the new Gmail" pop-ups).