If x comments in a post move post to category y

You could use the wp_insert_comment action to detect when new comments are added to a post. Then on your action callback, you’d the post’s comment count with get_comments_number(), and change post categories with wp_set_object_terms().

Something along these lines,

add_filter(
    'wp_insert_comment',
    function($id, $comment) {
        if ( (int) $comment->comment_approved ) {
            $post_id = (int) $comment->comment_post_ID;
            $post_comment_count = get_comments_number(
                $post_id
            );

            if ( $post_comment_count > 3 ) {
                wp_set_object_terms(
                    $post_id, 
                    array('3-comments'), 
                    'category', 
                    false // don't append, overwrite terms
                );
            } else if ( $post_comment_count === 2 ) {
                wp_set_object_terms(
                    $post_id, 
                    array('2-comments'), 
                    'category', 
                    false // don't append, overwrite terms
                );
            } else {
                wp_set_object_terms(
                    $post_id, 
                    array('1-comment'), 
                    'category', 
                    false // don't append, overwrite terms
                );
            }
        }
    },
    10,
    2
);