[Vuejs]-Push is not a function vuejs

i don’t sure if this will resolve your problem but is unnecessary do:

<button class="btn btn-primary" @click="addProductToCart(product)">Add to cart</button>

because you have product as prop of the component, should be @click="addProductToCart" without problem.
and your method should be so:

addProductToCart() {
  axios
    .post('/products/create', {
      product: this.product,
    })
    .then(response => {
      console.log(response);
      this.$emit('addedToCart', this.product);
    });
}

One thing more, use kebab-case to call the key string when you emit to the parent component:

this.$emit('addedToCart', this.product);

replace it with:

this.$emit('added-to-cart', this.product);

Then in your parent component you have:

<Productlist :product="product" @addedToCart="addedToCart"></Productlist>

replace it with:

<Productlist :product="product" @added-to-cart="addedToCart"></Productlist>

I guess this last things will resolve your problem according the Vue documentation.