Change term name only on front

You can detect if you are in the frontend using the function is_admin(), like so:

add_filter('get_term', 'filter_term_name', 10, 2);

function filter_term_name($term, $taxonomy) {
    // return the term untuched if you are in the admin
    if( is_admin() ) {
        return $term;
    }
    if ($taxonomy == 'my-taxonomy') {
        // numeric value of term in another taxonomy
        $meta_value = get_term_meta($term->term_id, '_my_meta_name', true);

        if ($meta_value) {
            // get name of meta value from another taxonomy
            $value = get_term($meta_value, 'another-taxonomy');

            if ($value) {
                $term->name = sprintf('%s, %s', $value->name, $term->name);
            }
        }
    }

    return $term;
}