It seems like the real main problem people have with image-based identification questions is that many (though not all) of them can be solved by reverse image searching, a tool which is by now well-known here (though not more broadly on the internet). There are several points that I'd like to make regarding this specifically.
- We've had a warning appear for new identification-request questions for almost 6 months. In that time, the fraction of such questions has, by my estimate, dropped. I don't have solid statistics on this, but from looking at the most recent 100 questions in the tag, it looks like only about 20% of them are of this type.
- The problem with these questions isn't a problem with any individual question, but rather with them all being highly similar. Users here get bored of the same answer over and over again. If we got only one or two questions of this type a year, it's hard to believe anyone would complain. Nor is anyone complaining much about the questions on which reverse image search fails, but which can be solved in other ways.
If this is really a problem (which I'm not sure it is, but other people are apparently convinced of it), rather than proposing a new rule (which will make things more confusing and unwelcoming for new users) I suggest we take the approach that other SE sites adopt when they frequently get slight variants of the same question over and over, specifically, making a canonical question and marking duplicates. New moderation rules add complexity to the site (both for new users and the high-rep users who enforce them) and tend to cause problems even for some otherwise fine posts, so they should be avoided whenever alternative solutions exist. Marking duplicates rather than adding a new requirement is more philosophically sensible, since the problem is really that all these questions are, in some sense, abstract duplicates.
We should write a canonical question/answer showing how to identify a picture with Google reverse image search. In fact, we already have such a tutorial, but it's on meta rather than the main site. Once we post the canonical question, future image-based identification requests which can be answered via a Google search should be marked as duplicates of the canonical post. Of course, we should be certain that Google actually does answer the question, which means checking the results. If most or all of the top results point to the answer (in English) then it is a duplicate. If there are a lot of irrelevant results at the top, or the answer doesn't even appear until going through a couple pages, you should not assume your search results will be the same as the asker's, so it's better just to answer the question.
The above addresses what seems to be the primary "con" point in the OP and which is also the main complaint of most people with regard to these questions. For completeness, I'll briefly state why the other points aren't such big problems.
Con: They are completely SEO-unfriendly; that is, no one can benefit from them by searching Google, even if they are looking for the same image's source.
This is only true if your definition of "search" in SEO is "Google text-based search". But if we're really all about promoting image searches, we should really include image searches in the definition of "search". And in those terms, these questions do pretty well. They do especially well in the case where reverse image search doesn't already know what the image is, which incidentally, is the important case that will not be marked as duplicates under my proposal.
Sure, they won't be bringing in tons of traffic that way. But our goal here is decidedly not to simply maximize traffic. It's to have the best answers for anime and manga questions, whether they're common or uncommon. If the origin of a particular image is one such question (and not already answered in many other places), we want to be at the top of that image search, just the same that we want to be at the top of an ordinary text-based searches.
Con: We are frequently asked about the source of images which are fan art or completely original characters. (Or vocaloid mascots.)
I don't see a problem with fan art if we can identify it. The appeal of a piece of artwork isn't solely in the art style. Even if we could a priori restrict fanart, I wouldn't want to. The same is true for original characters. This identification request (which, incidentally, was not answered by a Google search at the time it was asked) is about an original character. But despite that, there's plenty of other images of the same character, and even some short manga.
To put it simply, I don't see why the popularity or amount of related content should factor into our consideration at all from a moderation standpoint. It already gets factored in from a voting standpoint, as more people will vote for things they know about, but moderation ought to be independent of such considerations. Whether the source is fanart of an original character by a no-name artist or from a major series, it should be just as on-topic and welcome here.
Con: The size and quality of these images is frequently questionable, at best.
Con: (Related to above) They are frequently avatars that are cropped to the point where the character in them could be one of many, especially if it's fan art.
I've grouped these together because they're really the same point. I've only seen a couple images like this ever. It's not "frequent" by any stretch. Only one of them was bad enough that one could really not even tell that the image was anime-related or what it was supposed to be, and that one was quickly dealt with by the existing policy. The other one was a low resolution version of a large image (maybe 50x50), but in that case we actually managed to find the right answer despite the low quality. I no longer have the link handy for either, but having looked through some more questions recently, I haven't seen any more examples.
So, out of a hundred or more questions that I've seen like this, there have been only a couple cases which were described by this, and only one in which that was an issue big enough to make the question unanswerable. To put it simply, this is not a "frequent" problem, it's an outlier, and most certainly not something to base site-wide policy off of. We'll always have outliers; the goal of policy is just to create a system which will work well for most of the questions most of the time, and there's always going to be some element of subjective decision with regard to the exact boundaries. So far, this has not proven a problem (we've handled every case of it without any issues), so there's really no need to debate it here further.