JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/select2/4.0.13/css/select2.min.css">
<script src="https://cdnjs.cloudflare.com/ajax/libs/select2/4.0.13/js/select2.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery-validate/1.19.3/jquery.validate.min.js"></script>
   <form action="#" name='test'>
     <div class="reg__grid">
       <input type="text" name="name" placeholder="nameііі">
     </div>

     <div class="block__wrap">

       <input id="b_1" type="radio" checked class="block__input" name="place">
       <label for="b_1" class="block__label">
         <span class="block__title">У отделение</span>
       </label>
       <div class="block__box">
         <select class="city__select" name="department" data-placeholder="Выберете отделение" data-search="Поиск">
           <option></option>
           <option>отделение № 1 ул. Старо 11</option>
           <option>отделение № 2 ул. Старо 11</option>
           <option>отделение № 3 ул. Старо 11</option>
           <option>отделение № 4 ул. Старо 11</option>
         </select>
       </div>

       <input id="b_2" type="radio" class="block__input" name="place">
       <label for="b_2" class="block__label block__label--inner">
         <span class="block__title">На почту</span>
       </label>
       <div class="block__box">
         <select class="city__select" name="department" data-placeholder="Выберите почтомат" data-search="Поиск">
           <option></option>
           <option>Поштомат № 1 ул. Старо 11</option>
           <option>Поштомат № 2 ул. Стар 11</option>
           <option>Поштомат № 3 ул. Старо 11</option>
           <option>Поштомат № 4 ул. Старо 11</option>
         </select>
       </div>
     </div>
     <button type="submit" class="reg__next">Дальше </button>
   </form>

CSS

.block__box {
  margin-bottom: 25px;
}

input {
  outline: none;
  margin-bottom: 5px;
}

input.error {
  border-color: red;
}

JavaScript

$(document).ready(function() {

      $('select[name="department"]').select2({
        placeholder: $(this).data('placeholder'),
        language: "uk"
      }).on('select2:open', function(e) {
        let textPlaceholder = $(this).data('search');
        $('.select2-search input').prop('focus', false);
        $(this).data('select2').$dropdown.find(':input.select2-search__field').attr('placeholder', textPlaceholder);
      });
      
      // validation

      $('form').validate({
        rules: {
          name: {
            required: true,
          },
          department: {
            required: true,
          }
        },
        messages: {
          name: "Поле обязальное",
        },

      });
    })