[Vuejs]-Getting the dropdown to show based on an ID in vue

You forgot to add the v-model to the select in the second page. You can find some example here.
Your code should be something like:

<select id="product_id_select" v-model="selectedProductId" class="form-control select2" style="width: 100%;">

where selectedProductId is a value in your data (or a computed property) that contains the value inserted into the local storage in the first page