What is the text that appears beside the page titles in the list of pages in the WP admin?

It’s called post state, but not to be confused with post status (publish, draft, etc.) (although the state can of course contain the post status), and the core WordPress function which adds the post states such as the Privacy Policy Page to the posts (Posts, Pages, CPTs) list table is _post_states(), but the states are retrieved using get_post_states() which fires the display_post_states filter that you can use to add your own post states.

So for example, this adds a Foo State to the post where the ID is 123:

add_filter( 'display_post_states', 'my_post_states', 10, 2 );
function my_post_states( $post_states, $post ) {
    if ( 123 === $post->ID ) {
        $post_states['foo_state'] = 'Foo State';
    }

    return $post_states;
}

Update

In reply to your 1st comment:

Where are these post states saved in the database?

Note that a “post state” is really just a specific condition that the post is currently in, just like the “Privacy Policy” page which is the current privacy policy page on the site in question. So its current state/condition is, well, that it is the privacy policy page.

And if later on you created a new page and set it as the privacy policy page, then that new page would be the one having the Privacy Policy Page state. Likewise, if you changed your WooCommerce settings where you chose a new page as the cart page, then in the Pages list table, the newly selected cart page would then have the Cart Page state instead of the old cart page.

And if for example you have a custom login page, you can add a state labeled Login Page to the post in the (Pages) list table so that the user knows that it is the current custom login page, just like what WooCommerce did with the cart, checkout and “My Account” pages.

So in the above cases, the post state itself is not saved in the database, but they are formed based on the post data that are in the database, e.g. the post ID, type, status, etc. (and custom options in the wp_options table, a post meta in the wp_postmeta table, etc.)

And there are other default/core post states that you can check here, but two of them are:

• Front Page — this state is added if your homepage is set to a static front page (post type page)

• Posts Page — this state is added if your “blog”/posts page is set to a custom Page (post type page) — see the above link for more information

In reply to your 2nd comment:

Are post states meant to be shown only within the WP admin area, or
can they be shown in the public facing parts of the site?

It would be the former (intended for admin use), but you can display the post states in the public facing parts, if you want to — WordPress didn’t forbid you to display the post states outside the admin area. However, the post state functions are defined in wp-admin/includes/template.php, so you’d need to manually load it.