If an identification question has an accepted answer, it would make sense to tag it with the author's name. Or should we not tag it and leave it as a sort of trivia question for others?
No, I don't think so. Not all story identification questions are resolved unambiguously, and sometimes the asker misremembered details. If you know the name of the author already, the question is often not interesting.
The tags on a question are tags that searchers might use. By definition, someone searching for an identification question does not know the name of the author. To retag the question would be counter-productive.
I would encourage using tags that reflect what themes the asker remembers, e.g. What is the title of a story about a elevator through time and the smallest possible change → time-travel obviously, and parallel-universe (I'm not sure about that one — I put it in because it involves changing the timeline, but I can see the case against).
I'm going to say that yes, providing that the answer is clearly proved to be the correct one (by an answer being accepted, for example). If there is any doubt, the the author's name should be excluded, until all doubt is removed.
It's a nice idea, but where there isn't a tag in place for the author remember that (it seems to me that) quite a few of the story-identification questions are raised by first-time or relatively new visitors who may not have or indeed ever gain the rep required to create a new tag.
Current standard practice or not, yes, I'd like to see this happen. My argument for is that if I subscribe to an author's tag, it's because I am interested in seeing discussion about the author and their works. I'd like to know what others are saying about them, what is sticking out to other readers (what one person remembers might be different from what I remembered; what struck them powerfully might be different from my own interaction with the work). I'm following an author tag because I am a fan, and even though an answer might be accepted, if I am a knowledgeable fan of that author, I might be able to contribute to the question/answer through comments or provide an even better answer :)
I think it would be a waste of time to add these tags. Single use tags without a tag wiki get deleted automatically by the system eventually (see Do not expire single-use tags that have a tag wiki), so we would just end up writing a whole lot of unnecessary tag wikis or else have things deleted. Plus, after all this time of not doing it this way, it would be too much work to change.
Instead of relying on tags, use an advanced search:
[story-identification] isaccepted:yes is:a + title of the work.
You can tweak this search to remove the requirement that the answer be accepted or search for just an author instead of a title. It's very flexible.
Feel free to add more information to answers (accepted or not) to make it easier to find them. (This is especially important when the title of the work is generic.) For example, it might be good to add the full name of the author, the year of release, or a description to the answer.
One reason tags exist is not for people searching for answers, but people searching for questions. People who follow an author's tag are going to have interest in story-identification about their works.
Note that this is beneficial even if the tag were placed incorrectly, in fact particularly so, because an expert on that author's works is more likely to recognize the mistake than others would. And of course they may also help to confirm or add weight via comments or additional answers when they believe it is correct.
I'll say Yes, because it would help in ordering the information after the question has been answered, as Dima pointed out.
Also, it might be an interesting metric later for figuring out what author has the least memorable titles.
I previously tried to do this with my answer here (it was not a short story, but a book), but my edits were rejected by @Bellatrix. I, however, do believe that we should tag the question with the author and whatnot because if anyone else wants to peruse the site, we want them to get the answers they were looking for.