JSFiddle - React, Tailwind, and code Playground

HTML

<form action="/search-results" method="get" id="views-exposed-form-home-search-block" accept-charset="UTF-8"><div><div class="views-exposed-form">
  <div class="views-exposed-widgets clearfix">
          <div id="edit-type-wrapper" class="views-exposed-widget views-widget-filter-type">
                  <label for="edit-type">
            Class          </label>
                        <div class="views-widget">
          <div class="form-item form-type-select form-item-type">
 <select id="edit-type" name="type" class="form-select"><option value="All" selected="selected">- Any -</option><option value="bull">Bulls</option><option value="cowpair">Cow/Calf Pair</option><option value="cow">Cows</option><option value="feeder">Feeders</option><option value="female">Replacement Females</option><option value="stocker">Stockers</option></select>
</div>
        </div>
              </div>
          <div id="edit-field-class-tax-tid-wrapper" class="views-exposed-widget views-widget-filter-field_class_tax_tid">
                  <label for="edit-field-class-tax-tid">
            Hidden          </label>
                        <div class="views-widget">
          <div class="form-item form-type-select form-item-field-class-tax-tid">
 <select id="edit-field-class-tax-tid" name="field_class_tax_tid" class="form-select"><option value="All" selected="selected">- Any -</option><option value="139">Stockers</option><option value="136">Cow/Calf Pairs</option><option value="134">Cows</option><option value="138">Feeders</option><option value="135">Replacement Females</option><option value="137">Bulls</option></select>
</div>
        </div>
              </div>
          <div id="edit-field-breed-category-tid-wrapper" class="views-exposed-widget views-widget-filter-field_breed_category_tid">
                  <label for="edit-field-breed-category-tid">
            Breeds          </label>
                        <div class="views-widget">
          <div class="form-item...

JavaScript

$("#edit-type").change(function(){
    $("#edit-field-class-tax-tid").val($(this).val());
});