While certain Android device manufacturers have enabled system-wide search, it is not included in the core OS. That is set to change with Android 12, since beta 4 for Pixel phones now supports it, despite the fact that the feature has not yet been formally published. It appears that construction on it is still ongoing, as it appears to be unfinished.
The search bar in the app drawer is the only way to get to it right now; the search widget on the homescreen still uses the previous search feature. A handy new option for the app drawer can make the keyboard appear every time you open it, allowing you to immediately start typing to find what you’re looking for.
Shortcuts (for applications and people), persons (contacts, but also discussions with them), settings (including toggles and other controls), and tips are all searchable using the on-device system-wide search. If you don’t want a category to appear in the results, you can disable it separately.
Google is reportedly testing the addition of even more search categories, but it does not appear to be ready for prime time.