Page that ‘subscribes’ to multiple categories

Enable the category taxonomy for your custom post type, or enable it for the default page post type:

function wpa_cats_for_pages(){
    register_taxonomy_for_object_type( 'category', 'page' );
}
add_action( 'init', 'wpa_cats_for_pages' );

Then within your page template, get the category IDs assigned to the client page and create a new query which passes those IDs:

// get IDs of categories assigned to this client page
$categories = wp_get_object_terms( $post->ID, 'category', array( 'fields' => 'ids' ) );

// query for posts with categories that match those assigned to this page
$args = array(
    'posts_per_page' => -1, // get all matching posts
    'category__in' => $categories
);

$client_posts = new WP_Query( $args );

// output post data
while( $client_posts->have_posts() ):
    $client_posts->the_post();
    the_title();
endwhile;

// reset the global $post object so template tags for this page work as expected
wp_reset_postdata();

Then for each client page, you’ll just select the categories of posts you want displayed on that page, the same way you assign categories to the posts.