With Stage Manager enabled, you will see the new Windowed UI (User Interface) - and with this you will have different controls for the windows. The corner “handle” that you describe is used to resize the related App window.
With Stage Manager disabled (arguably, this being the more useful mode when using the iPad without a connected external monitor), you will see the established Full, Split and Slide-over views listed and accessible from the three-dot button. In this mode, you will not see the handles that can resize the App windows.
Stage Manager Multitasking views can be selected or disabled/enabled from iPad settings:
Settings > Multitasking Gestures > [Multitasking] - select Split View & Slide Over or OFF
You may find that the Split View & Slide Over mode is more to your liking - allowing use of multitasking features, but without the new windowed presentation of Stage Manager. The Stage Manager feature can also be simply enabled/disabled from the iPad Control Centre.
Move, resize, and organize windows with Stage Manager on iPad - Apple Support