This is the screen you’re referring to:
That interface is a modal dialog created using javascript. Specifically, backbone.js. Internally WordPress stores files in the file system and creates a post for them of type attachment. This attachment post stores the title, caption and other information. Because of this, it’s possible to have comments on an image, categories etc, in the same way that posts of type page and post do. In WordPress, the attachment posts ID is stored, not the file itself, e.g. a featured image is an attachment ID in post meta. This way we don’t need multiple copies if the image appears in multiple places.
This means there is a full post edit screen for any image.
However because we have this screen, and it’s powered by javascript, when you edit the fields, they’re automatically saved. There is no need for an update/save button. As soon as you modify any field, it sends the update once the field looses focus or the enter key is pressed, showing a spinner then the word ‘saved’ in the corner:
Clicking on edit more details also shows a full attachment post edit page with all of those fields and more, including an update button:
If you really wanted to, you could add a button to submit the form using javascript, however, the code is undocumented, you’d need to disable the auto-saving, and the whole thing might be changing in the future