You have a few problems with the code.
First problem in your code is the $(element).text()jquery text() strips your code from html including the <br> tags. So there is no newlines in the clipboard text since all html newlines are stripped away.. so nothing to replace.
If you want to keep <br> as newlines you need to use .html() and parse your text more manually.
Second problem is that you use <input> tag. <input> tag is only single lines so u cant have any newline in there. you need to use <textarea> for the conversion.
The last problem is as above that you also should use \r\n for windows users.
I updated your snippet with a working version.
function copyToClipboard(element) {
var $temp = $("<textarea>");
var brRegex = /<br\s*[\/]?>/gi;
$("body").append($temp);
$temp.val($(element).html().replace(brRegex, "\r\n")).select();
document.execCommand("copy");
$temp.remove();
}
$( "#FailCopy" ).click(function() {
alert("Well done! div #error-details has been copy to your clipboard, now paste it in the notepad or email!");
});<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.2.1/jquery.min.js"></script>
<!--text that i want to copy-->
<h2>Div #error-details: the text I want to copy to clipboard:</h2>
<er id="error-details">Name: test<br>Surname: test<br>Email: [email protected]<br>Address: test<br>City: test<br>Country: null<br>Ad Category: test<br>Plan: null<br>Website: <br>Company name: test<br>Μήνυμα: test</er>
<br><br>
<button id="FailCopy" type="button"
onclick="copyToClipboard('er#error-details')">Copy div to clipboard</button>