Add dynamic color to each category item

You are getting the color for one category (term is more appropriate though)

$the_category_id = $category[0]->cat_ID;

instead of checking the ID and Color for each term, that’s why it’s applying the same to all.

Try this (untested):

<?php

    $categories = get_the_category();
    $sep = '';
    foreach ($categories as $cat) {
        $the_category_id = $cat->term_id;
        if(function_exists('rl_color')){
           $rl_category_color = rl_color($the_category_id);
        } else {
           $rl_category_color="#000"; // maybe a default color?
        }
        echo $sep . '<a href="' . get_category_link($cat->term_id) . '"  class="' . $cat->slug . '" title="View all posts in '. esc_attr($cat->name) . '" style="background-color:' . $rl_category_color . '">' . $cat->cat_name . '</a>';
        $sep = ', ';
    }

?>