Getting custom posts by post id from cutomizer text input

The post__in parameter needs an array, see the WP_Query docs.

With your customizer field you are not saving an array, you are saving a comma separated string. So try something like this:

$my_field = get_theme_mod('fp_post');

// create an array from comma separated values
$the_post_id_array = explode(',', $my_field);

$query = new WP_Query( array( 'post_type' => 'post', 'post__in' => $the_post_id_array ) );

while ($query -> have_posts()) : $query -> the_post();

    the_title();

endwhile;

See in the docs:

'post__not_in' => array( '1,2,3' )  // <--- this wont work