Limit 1 post by user in each custom post type

You’re using wp_count_posts( for counting posts, so it will return the number of posts.

If you want to get the number of posts for given user, you should use count_user_posts instead.

So change this line:

$count_post = wp_count_posts($post_type_name)->publish;

to:

 $count_post = count_user_posts( get_current_user_id(), $post_type_name, true );