Setting a new / override shippingAddress via axios
Stumbled over the old questions, this is how it works:

updateShipping() {
        const options = {
            method: 'post',
            url: '/actions/commerce/cart/update-cart',
            headers: {
                'X-CSRF-Token': window.Craft.csrfToken,
            },
            data: qs.stringify({
                shippingMethodHandle: 'standardShipping',
                shippingSameAsBilling: 0,
                shippingAddressId: 123, // remove if it doesn't exists or is new
                shippingAddress: {
                   id: '', // only needed if address is new
                   firstName: 'test',
                   ...
                },
            }),
        };
        axios(options)
            .then(response =&gt; {...})
            .catch(error =&gt; {...});
    },