Looks like it’s only Pixel phones that have this feature for now:
We’re seeing Screen search rolled out on several Pixel phones running the latest Google app beta (version 14.31). It’s not yet live on all devices, and Google previously said it would roll out globally over the coming months.
Just installed the beta of the Google app on my Pixel 7a and I still don’t have that “Search screen” feature in the Google Assistant panel. Maybe others have more luck with their phones?