WordPress provides a get_search_form filter using which the *all* the search forms' markup can be modified. But it does not seem possible at this time to do the changes for specific instances of search forms by targetting their ID or any other kind of identifier.
So then what can be done for now is to simply place the HTML & PHP of the markup and modify as needed.
Here's the code to display a stock unmodified or default search form in HTML5 output:
