Post Type only showing for “Admin” under “Appearance” > “Menus”

Can you see it when you open the ‘Screen Options‘ menu on the top right of the admin next to the ‘Help‘ button? And with another user name?

Am a bit confused because your code doesnt show any register_post_type() function.
Also, do you want to create a custom post type or a custom taxonomy?

Try with this basic code to create a custom post type, to see if it works for you:

/* books custom post type */
add_action( 'init', 'create_books' );
function create_books() {
    register_post_type( 'cpt-books',
        array(
            'labels' => array(
                'name' => __( 'Books' ),
                'singular_name' => __( 'Book' )
                ),
            'public' => true,
            'show_ui' => true,
            'publicly_queryable' => true,
            'exclude_from_search' => false,
            'menu_position' => 4,
            'query_var' => true,
            'supports' => array( 'title', 'editor' )

        )
    );
}