I've tried to put the custom control inside the view panel, thus each value in the column with context menu has its own instance of the dialog and ... it simply works, the dialog title show the expected information.

Unfortunately there is still BUT. It does not work for cases when the view is categorized. In this case the menu is not displayed properly, it appears at the end of the page with no or at least not expected formatting. The menu items work as expected but it is cannot be used.