JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="http://code.jquery.com/ui/1.8.18/themes/base/jquery-ui.css">
<label for="monthDate1">NOT pointing to previously selected date:</label>
<input id="monthDate1" type="text" class="date-picker" />
<br/>
<br/>
<label for="monthDate2">Pointing to previously selected date:</label>
<input id="monthDate2" type="text" class="date-picker" />

JavaScript

$(function() { 
//set datepicker or month/year selection
        $('#monthDate1').datepicker( {
            changeMonth: true,
            changeYear: true,
            showButtonPanel: true,
            dateFormat: 'MM yy'
        }); 
        //needed to hide calendar with days
        //this bit is adopted from https://gist.github.com/877276#file_gistfile2.html
        $("#monthDate1").focus(function () {
            $(".ui-datepicker-calendar").hide();
            //add event to perform on done button click
            $(".ui-datepicker-close").click(function(){
                var month = $("#ui-datepicker-div .ui-datepicker-month :selected").val();
                var year = $("#ui-datepicker-div .ui-datepicker-year :selected").val();
                $("#monthDate1").datepicker('setDate', new Date(year, month, 1));
            });
        });
    
        $('#monthDate2').datepicker( {
            changeMonth: true,
            changeYear: true,
            showButtonPanel: true,
            dateFormat: 'dd/mm/yy'
        });
        $("#monthDate2").focus(function () {
            $(".ui-datepicker-calendar").hide();
            $(".ui-datepicker-close").click(function(){
                var month = $("#ui-datepicker-div .ui-datepicker-month :selected").val();
                var year = $("#ui-datepicker-div .ui-datepicker-year :selected").val();
                $("#monthDate2").datepicker('setDate', new Date(year, month, 1));
            });
        });
});//end of jQuery.ready()