JSFiddle - React, Tailwind, and code Playground

HTML

<div class="custom-selectbox">
    <div class="inner"><span>--</span></div>
    <select name="month">
        <option value="" selected="selected">--</option>
        <option value="1">January</option>
        <option value="2">February</option>
        <option value="3">March</option>
    </select>
</div>

<div class="custom-selectbox">
    <div class="inner"><span>--</span></div>
    <select name="month">
        <option value="" selected="selected">--</option>
        <option value="1">January</option>
        <option value="2">February</option>
        <option value="3">March</option>
    </select>
</div>

CSS

.custom-selectbox {
    position: relative;
}
    .custom-selectbox select {
        position: absolute;
        top: 0;
        left: 0;
        z-index: 1;
        width: 240px;
        height: 36px;
        opacity: 0;
    }
    .custom-selectbox .inner {
        position: relative;
        z-index: 0;
        box-sizing: border-box;
        border: 1px solid #000;
        padding: 0 12px;
        width: 240px;
        height: 36px;
        font-size: 14px;
        line-height: 36px;
    }
    .custom-selectbox .inner:before,
    .custom-selectbox .inner:after {
        content: '';
        position: absolute;
        display: block;
    }
    .custom-selectbox .inner:before {
        top: 0;
        right: 0;
        bottom: 0;
        width: 24px;
        background: #000;
    }
    .custom-selectbox .inner:after {
        top: 50%;
        right: 7px;
        margin-top: -3px;
        width: 0;
        height: 0;
        border: solid transparent;
        border-top-color: #fff;
        border-width: 5px 5px 0 5px;
    }

JavaScript

function change_selectbox(select) {
	var item = select.children('option:selected').text();
    select.prev('.inner').find('span').text(item);    
}

jQuery(function(){
    // セレクトボックスの変更時に.innerのspanを書き換え
    jQuery('select').change(function() {
        change_selectbox(jQuery(this));
    });
});