Vue

HTML

<div id="app">
  <div 
                          
                            <a href="#" class="uk-icon-button" @mouseover.stop="showDrop = true">
                                Иконка
                            </a>
                            <div class="dropdown uk-box-shadow-medium" v-show="showDrop" @mouseout="showDrop = false">
                                <div class="auth" v-show="showAuth">
                                    <div class="uk-card">
                                        <div class="auth-title">
                                            <span class="uk-text-bold">Авторизация</span>
                                        </div>
                                        <div class="auth-form"><input type="text"></div>
                                        <div class="auth-form auth-form-reg">
                                            <span class="uk-text">Вы еще не с нами?:)</span>
                                            <br>
                                            <a @click="onClick" href="#" class="uk-link auth-form-reg-link">Зарегистрироваться!</a>
                                        </div>
                                    </div>
                                </div>
                                <div class="auth" v-show="showReg">
                                    <div class="uk-card">
                                        <div class="auth-title">
                                            <span class="uk-text-bold">Регистрация</span>
                                        </div>
                                        
                                        <div class="auth-form auth-form-reg">
                                            <span class="uk-text">Есть аккаунт?</span>
                                            <br>
                                            <a @click="onClick" href="#" class="uk-link auth-form-reg-link">Авторизоваться!</a>
                                        </div>
 ...

Vue

new Vue({
  el: "#app",
  data:{
                showDrop:false,
                showAuth: true,
                showReg: false
            },
            methods:{
                onClick: function(){
                    this.showAuth = !this.showAuth
                    this.showReg = !this.showReg
                }
            }
})