How to rename the add post/page sidebar panel sections in wordpress without plugin?

You can change the category name using the init action

function revcon_change_cat_object() {
    global $wp_taxonomies;
    $labels = &$wp_taxonomies['category']->labels;
    $labels->name = 'category_new_label';
    $labels->singular_name = 'category_new_label';
    $labels->add_new = 'Add category_new_label';
    $labels->add_new_item = 'Add category_new_label';
    $labels->edit_item = 'Edit category_new_label';
    $labels->new_item = 'category_new_label';
    $labels->view_item = 'View category_new_label';
    $labels->search_items = 'Search category_new_labels';
    $labels->not_found = 'No category_new_labels found';
    $labels->not_found_in_trash = 'No category_new_labels found in Trash';
    $labels->all_items = 'All category_new_labels';
    $labels->menu_name = 'category_new_label';
    $labels->name_admin_bar = 'category_new_label';
}
add_action( 'init', 'revcon_change_cat_object' );

you can also change the tag labels like.. change this line

$labels = &$wp_taxonomies['post_tag']->labels;