Floating

Your reference site has this in its CSS code that displays the "new" you're seeing:

li#menu-item-[menu item id] a:after {
    content: "new";
    position: absolute;
    font-size: 11px;
    font-weight: 600;
    top: 12px;
    line-height: 1;
    color: #fff;
    background-color: #ff763d;
    padding: .15rem .25rem .125rem .25rem;
    right: 0;
    letter-spacing: 1px;
    -webkit-border-radius: 2px;
    -moz-border-radius: 2px;
    border-radius: 2px;
}

You can apply this on any link, even on the footer ones as long as you know the CSS selector.

But if you want this to be dynamic, depending on any new updates, that'll require site customization that is outside of our scope.