Background process under Monterey

Are you closing a window by pressing the red button?

Unlike in Windows, Mac applications are not quit when you close a window (*)

Any application that can have multiple windows (like, for example, Safari, Word, Preview, ...) is NOT quit by closing a window (even if it is the only one at the moment). That should be quite evident since the name of the application will remain in the menu bar, indicating that it is still running and in the front.

The quickest way to quit an application is to press the default shortcut, Command-Q.

You can also choose "Quit" from the application menu - for example:

(*) there are a few exceptions, comprised of "single window" applications - applications that can only have one window; these are indeed quit when closing that single window. (Example: System Preferences)

In that case, what indication do you have that the applications are still running after you quit them?

Do you see the tiny dot under tgem in the Dock? Do you see them in Activity Monitor?

Picking up on previous very good explanation from @ Luis S above.

Refer to below image

Some Applications like Photos and System Preference will close Both the Active Window of the Application as well as Quit.

Others do not

What is “this app”?

There are literally hundreds of apps running in the background at any time. This is normal.