In order to copy the content of the WebBrowser control to the clipboard as Text you need to dive into the underlying COM interfaces inside the webbrowser control. IHTMLDocument2 gives us the selection. We just need to pull the text out of it:
IHTMLDocument2^ htmlDocument = (IHTMLDocument2^) webBrowser1->Document->DomDocument;
IHTMLSelectionObject^ mySelection = htmlDocument->selection;
String^ text = ((IHTMLTxtRange^ )mySelection->createRange())->text;
Clipboard::SetText(text);