Session not storing data in Vue.js & Laravel

I am trying to store data into the Laravel session on each AJAX call from the vue.js.

I want to add a product in Laravel cart using session but it's not storing any kind of session data. I have also added web middleware but no luck.

api.php

<?php
    Route::group(['namespace' => 'Api'],function(){
        Route::resource('product','ProductController');
        Route::post('add-to-cart','ProductController@addToCart');
    });

ProductController.php

<?php
    public function addToCart(Request $request){
    $product = $request->all();

    $cart = Session::get('cart');
    $cart[$product['id']] = $product;
    Session::push('cart', $cart);

    return Response::json(['success' => true, 'cart_items' => Session::get('cart')]);
}

ProductListComponene.vue

addToCartProduct(product){
    fetch('api/add-to-cart',{
        method: 'post',
        body:JSON.stringify(product),
        headers:{
            'content-type':'application/json'
        }
    })
    .then(res => res.json())
    .then(res => {
        console.log(res);
    })
    .catch(err => console.log(err));
}