How can we print out total number of posts filtered by post type and category?

If I’m understood, you want to run a loop and print the count for every term of your post. You can do so by this code:

foreach((get_the_terms($post->ID, 'project_category')) as $term) { 
    $term_object = get_term( $term->term_id, 'category' );
    echo $term_object->count;
}

Which will do what you asked for.