Custom Post Types in plugins?

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Yes, there are lot of plugins using CPTs as data holders (e.g., Advanced Custom Field). Others, though, carefully use the class WP_List_Table to conform to the default interface (e.g., BackWPup).

When registering a post type, we have options to control its visibility and “querybility”: public, show_ui, publicly_queryable and others. This is how ACF registers itself:

register_post_type('acf', array(
    'labels' => $labels,
    'public' => false,
    'show_ui' => true,
    '_builtin' =>  false,
    'capability_type' => 'page',
    'hierarchical' => true,
    'rewrite' => false,
    'query_var' => "acf",
    'supports' => array(
        'title',
    ),
    'show_in_menu'  => false,
));

So, yes, it’s perfectly fine to create a CPT that appears only in the backend, and then use it in other parts of the system. For example, the Navigation Menus (wp-admin/nav-menus.php) is a CPT, and registered like:

register_post_type( 'nav_menu_item', array(
    'labels' => $labels,
    'public' => false,
    '_builtin' => true, /* internal use only. don't use this when registering your own post type. */
    'hierarchical' => false,
    'rewrite' => false,
    'delete_with_user' => false,
    'query_var' => false,
) );

Your example, using as a Shortcode, would be something like this used in a Post or Page:

[my_shortcode id="77"]

and what it does is: get whatever information you have in your CPT with ID 77 and display it.

Forget functions.php for this kind of stuff. Research our archives on as there is plenty of material to get started, simple and complex.