Apart from the "story-identification" tag, shouldn't OPs mention the same in the question title so that it is easier to spot? I ask this because I see several questions with only the tag but nothing mentioned in the question.
Yes. Rather, the question should be all that is necessary to understand exactly what the OP is looking for.
Let's break down the three elements of a question post (as I see them).
The hook. The title should provide a good general overview of what the OP is looking for. It doesn't need to get into the nitty gritty details or criteria, but readers should have at least a clue of what the OP is asking for.
This should be 100% self sufficient. It should clearly explain what information the OP is looking for and lay out any criteria that should be considered (e.g. canon levels, universes, media, etc).
These should be used for categorization purposes. They should be helpful, but not necessary to the question itself.
For example, if I am looking for information about Wolverine's claws within the comic universe only, I would make sure that the question itself clearly states that I am looking for an answer based on the comic universe, and not the movie universe. I would also tag it wolverine and marvel-comics (I'd also understand the choice to use comics.)
In the case of story identification questions, the reader shouldn't need to look at the tags to understand that the OP is looking to identify a story; the tags should just be used to make it easier to categorize and find the question.
Yes you should indicate that you're trying to identify a story, but there's no need to repeat the tag wholesale.
Good: Trying to find 70s space opera with knights with magical powers and glowing swords.
Bad: Story identification: 70s space opera with knights with magical powers and glowing swords.
I agree that the body of the question should be completely self-contained, without reference to the tags.
However, I don't see why you're picking on the story-identification tag. Could you point to an example? I've read lots of story identification questions, and I've seen very few, if any, where it's not clear from the body of the question that an identification is wanted.
On the other hand, I've seen hundreds of silly plot-explanation questions where a tag is the only clue as to what work of fiction is being discussed; such as this one where the tag is the only hint that the question is about a certain popular kiddie-lit series from England. (Unless, of course, one happens to have read the books, and so recognizes some of the names, such as "Hogwarts".)