Removing Duplicate Value in Rest Api

What you can do is, find the unique values after getting results from your first REST call(After getting all possible employee names).

Then find the unique employees in your array and after getting unique values, append to dropdown. There are many ways to find the unique values in your array:

1. Using Plain Javascript.
2. Using JQuery.
3. Using plugins like lodash.js or underscore.js

You can find more details to get unique values from your array, here.

I see you have employee details in an arrays inside the results you get from SharePoint REST. For your case, you need to create a temporary array which will store the unique values of employee name. Check the code below:

var tempEmployeesArray = [];  

$.each(allEmployeeNames,function(i,employee){
  $.each(employee.Should_x0020_Cost_x0020_Modeler.results,function(j,item){
            if(employeeName==item.Title){
                if(tempEmployeesArray.indexOf(item.Title) < 0){
                    //code if duplicate not found
                    tempEmployeesArray.push(item.Title);
                    $("#EmployeeField").append("<option selected='selected' value='"+item.Title+"'>"+item.Title+"</option>");
                }

            }else{
                if(item.Title!=undefined){
                    if(tempEmployeesArray.indexOf(item.Title) < 0){
                        //code if duplicate not found
                        tempEmployeesArray.push(item.Title);
                        $("#EmployeeField").append("<option value='"+item.Title+"'>"+item.Title+"</option>");     
                }                        
                }           
            }   
        });

You incorrectly use your jquery ajax function getAllEmployeeNames. You shouldn't use async: false for the modern web. Correct one:

function getAllEmployeeNames() {
    return $.ajax({
        url: _spPageContextInfo.siteAbsoluteUrl + "/_api/web/lists/getbytitle('Should Cost Projects')/items?$select=Should_x0020_Cost_x0020_Modeler/Title&$expand=Should_x0020_Cost_x0020_Modeler/Id&$top=500",
        type: "GET",
        headers: {
            "Accept": "application/json;odata=verbose",
        }
    })
    .then(function(data){
        return data.d.results;
    })
    .fail(function(err){
        console.log(err);
        alert('error');
    });
}

then you can filter unique names using array (that one option):

getAllEmployeeNames()
    .then(function(results){
        var uniqueNames = [];

        results.forEach(function(employee){
            employee.Should_x0020_Cost_x0020_Modeler.results.forEach(function(name){
                if(uniqueNames.indexOf(name) === -1) {
                    uniqueNames.push(name)
                }
            })
        })

        // now you have array with unique names
        // add it to dropdown here
    })

Using the code below to remove the duplicates options of the dropdown list.

$("#EmployeeField option").each(function() {
        $(this).siblings('[value="'+ $(this).val() +'"]').remove();
});

Replace this part of code

$(allEmployeeNames).each(function() {
    $(this).siblings('[value="'+ $(this).val() +'"]').remove();
});

with

var uniqueEmployeeNames = [];
$(allEmployeeNames).each(employee.Should_x0020_Cost_x0020_Modeler.results, function(i, el){
    if($.inArray(el, uniqueNames) === -1) uniqueEmployeeNames.push(el);
});

May be below code will help you

First of all get all data and bind it to drop-down list as you can see below

This is HTML for reference

<div>
    <select id="ddlEmployee">
        <option value="0">select user</option>
    </select>
</div>

Below is Js Code

 function getDataREST(requestUrl, successFun, failFun){
   $.ajax({
      url: requestUrl,
      type: "GET",
      async:false,
      headers: {
          "Accept": "application/json;odata=verbose",
      },
      success: function (data) {
          successFun(data)
      },
      error: function (data) {
         failFun(data)
      }
   });
}
function bindEmployeetoDDL(){
    var addItemUrl = _spPageContextInfo.webAbsoluteUrl + "/_api/web/lists/getbytitle('UserLists')/items?$select=Id, Title,UserName/Id,UserName/Title&$expand=UserName&$top=2000" 
    getDataREST(addItemUrl, function (data){
        if(data && data.d.results.length > 0)
        {
            for(i=0;i<data.d.results.length;i++){
                 $("#ddlEmployee").append("<option value='"+data.d.results[i].UserName.Id+"'>"+data.d.results[i].UserName.Title+"</option>");
            }
            //removeDuplicateByValue()
            removeDuplicateByText()
        }
    }, function (error){
        console.log("Error while getting employee", Error)
    });
}

After successfully data bind to drop-down one of below method to remove duplicate entry from drop-down options

//This will be remove duplicate options by value
function removeDuplicateByValue()
{
    $("#ddlEmployee Option").each(function() {
        $(this).siblings('[value="'+ $(this).val() +'"]').remove();
    });
}

//This will be remove duplicate options by text
function removeDuplicateByText()
{
    $("#ddlEmployee Option").each(function() {
        $(this).siblings(':contains('+ $(this).text() +')').remove();
    });
}