Filter searches on the iOS App Store
No, there are not any hidden buttons or features to filter search results when searching from an iPhone.
It's done this way when viewing the App Store on iPadOS because iPhone apps can be run in iPadOS.  This is the default setting for developers using Xcode to create their apps.  Sometimes when searching, you just want to see apps that are designed only for iPads (developers determine if their app can be run on iPad only).
Because you can run an iPhone app on an iPad but not an iPad only app on an iPhone, Apple provides a way to search for only those types of apps.