How to exclude a category name from showing?

The function get_the_category is returning you an array of objects (see codex for more info), so if the post has more than one term assigned, then you are just outputting the first one.

So I think that doing a foreach loop will solve your problem:

foreach ($category as $cat) {
    if ($cat->slug != 'featured') {
        echo esc_html($cat->slug);
    }
}

Hope it helps!