If there is an ID question that has an established tag, should we edit and retag with the name of the story?
For the same reason as we shouldn't edit the answer into any question once it's solved. For story-ID questions, the story title is part of the answer, not the question. Editing it into the question, even in the form of a tag, would make people do a quick double-take and think "why did they need to ask when the correct answer is already right there?" before they check the revision history.
Tags are meant to aid searchability, but someone looking for questions about story X isn't going to care about reading an ID question whose answer turned out to be X.
For a question to be useful for posterity, it should generally be such that it would still be a valid question if asked today in the same form it's in now. Including the answer in the question defeats that purpose, even if it's only done later. Leave the answer in the answer, please.