How to translate functions like get_the_date()?

Why not do it in plain PHP?

For example, you can write a function like:

function get_the_german_date($date="", $post = null) {
    $d = get_the_date($date, $post);
    if ($d) {
        $d = new DateTime($d);
        return $d->format('d.m.Y');
    } else return false;
}

Similary, if you want to follow the WordPress standards:

function the_german_date($date="", $post = null) {
    $d = get_the_date($date, $post);
    if ($d) {
        $d = new DateTime($d);
        echo $d->format('d.m.Y');
    }
}

I realize this may be too much work since you want to do it dynamically depending on your locale, but you can do a check of your locale, or you can have a global date function where if website is in english you output get_the_date and if not the function above.