JSFiddle - React, Tailwind, and code Playground

JavaScript

var a = '"display, Name" <test@test.com>, display" Name <test@test.com>, test@test.com';
    var aa = a.match(/"(.*?)"(.*?),|(.*?),|(.*?)$/g); 
    aa.pop();
console.log(aa );

    var str = a;
    var insideQuote = false, arr = [], str1 = '';

    for(var i=0;i<str.length;i++) {
        switch(str.charAt(i)){
        
            case '"':
                str1 += str.charAt(i);
                insideQuote = !insideQuote;
                break;
          case ',':
                if(insideQuote ){
                    str1 +=  str.charAt(i);
                    break;
                }
                arr.push(str1);
                str1 = "";
                break;
            default: 
                str1 +=   str.charAt(i);
                
        }
        
    } 
    console.log(arr);