Removing user from Sharepoint User group api call succeds but user not getting deleted

The problem in your code is, you are referencing removeByLoginName but you are passing email ID to delete the user. so try the below code,

Deleting user by Email ID:

'use strict';  
var hostweburl;  
var appweburl;  
// Get the URLs for the app web the host web URL from the query string.  
$(document).ready(function()  
{  
    //Get the URI decoded URLs.  
    hostweburl = decodeURIComponent(getQueryStringParameter("SPHostUrl"));  
    appweburl = decodeURIComponent(getQueryStringParameter("SPAppWebUrl"));  
    // Resources are in URLs in the form:  
    // web_url/_layouts/15/resource  
    // Load the js file and continue to load the page with information about the folders.  
    // SP.RequestExecutor.js to make cross-domain requests  
    $.getScript(hostweburl + "/_layouts/15/SP.RequestExecutor.js", getuser);  
});  
function getuser()   
{  
    var executor;  
    var userEmail = "[email protected]";  
    // Initialize the RequestExecutor with the app web URL.  
    executor = new SP.RequestExecutor(appweburl);  
    executor.executeAsync({  
        url: appweburl + "/_api/SP.AppContextSite(@target)/web/sitegroups(6)/users/getbyemail('" + userEmail + "')?@target='" + hostweburl + "'",  
        method: "POST",  
        headers: {  
            "X-HTTP-Method": "DELETE"  
        },  
        success: function(data) {  
            alert("User Deleted successfully in SharePoint Group");  
        },  
        error: function(err) {  
            alert("error: " + JSON.stringify(err));  
        }  
    });  
}  
//Utilities  
// Retrieve a query string value.  
// For production purposes you may want to use a library to handle the query string.  
function getQueryStringParameter(paramToRetrieve) {  
    var params = document.URL.split("?")[1].split("&");  
    for (var i = 0; i < params.length; i = i + 1) {  
        var singleParam = params[i].split("=");  
        if (singleParam[0] == paramToRetrieve) return singleParam[1];  
    }  
}  

Deleting user by Login Name:

<script language="javascript" type="text/javascript" src="//ajax.googleapis.com/ajax/libs/jquery/1.8.1/jquery.min.js"></script>  
<script language="javascript" type="text/javascript">  
    $(document).ready(function() {  
        var removeUserUrl = "/_api/web/sitegroups/getbyname('Employee')/users/removeByLoginName";  
        var metadata = {  
            'loginName': 'i:0#.w|SharePointHOL\\KarthikJagan'  
        };  
        removeUser(removeUserUrl, metadata)  
    });  

    function removeUser(removeUserUrl, metadata) {  
        $.ajax({  
            url: _spPageContextInfo.webAbsoluteUrl + removeUserUrl,  
            type: "POST",  
            headers: {  
                "accept": "application/json;odata=verbose",  
                "X-RequestDigest": $("#__REQUESTDIGEST").val(),  
                "content-Type": "application/json;odata=verbose"  
            },  
            data: JSON.stringify(metadata),  
            success: function(data) {  
                console.log(data);  
                console.log("User has been successfully removed from the SharePoint Group.");  
            },  
            error: function(error) {  
                alert(JSON.stringify(error));  
            }  
        });  
    }  
</script>  

For more information, kindly have a look at the below links,

1. https://www.c-sharpcorner.com/article/add-and-remove-users-from-security-group-in-sharepoint-2016-using-rest-api/

2. https://social.msdn.microsoft.com/Forums/office/en-US/1c70bfb1-039d-43cd-94b5-6138893bde8b/rest-api-removedelete-sp-user-from-group?forum=sharepointdevelopment

3. https://www.c-sharpcorner.com/UploadFile/91b369/how-to-delete-users-from-the-group-in-sharepoint-using-rest/