Indispensable

With hundreds of posts, pages, testimonials, thousands of media, it is incredible that WordPress core doesn’t have such functionality.
Thank you Premio for saving me from insanity!

In media, you select the media and then click the organize button to add it to a given folder.

In the posts and pages regular menus, to move multiple pages at the same time, you select them, then in the bulk menu you select edit. A quick edit menu appears, you select the folder you want it added to. If you don’t want it to be attached to both current and destination folders, make sure to uncheck the one you do not want.