I actually have a few ideas on how to alleviate this problem, but I think to see most of them implemented would require effort from the part of Stack Exchange. Before I propose these ideas, I will cut to the chase and offer my opinion: it is feasible to keep identification request questions going as they are now, at least until this site leaves Beta. Granted, we need to be more stringent and add more QC guidelines and be more clear of what they are.
Option 1: Have a separate forum for identification requests
This could be a chat room or another Stack Exchange site reserved for identification requests. If such an SE site is formed, we could even merge Anime & Manga identification requests with those from Arqade and Movies & TVs, and just migrate all questions having only the identification-request tag to that site.
The current threshold to participate in the site's chat rooms is 20 reputations, which is not asking too much from the new users. We can even make this easier for them by rewarding reputations to users for completing the Tour. Now about the Tour: I really dislike it in its current unhelpful form. The site Tour should be more unique and tailored to Anime & Manga, and provide more incentives (e.g. privileges) for new users to complete them. More about this in the next section.
Option 2: Require new users to complete the Tour before they are able to ask identification request questions
When a user asks their first question, after they click on the Ask Question link, they should be brought to a page asking What is your question about? and a list of the 3–5 most popular tag names on the left and tag wikis on the right. The first in that list should be identification-request. If they check that, they will be brought to the updated Tour page, in which they will learn about some useful tools for identification request questions such as Google Reverse Image Search for images and Shazam for music. Then give them some tests similar in nature to the review tests (audits)review tests (audits) that could test them some identification questions that could be answered using Google Reverse Image Search to check that they know their tools. If they fail too many times, they will be banned.
Option 2.5: Further require some reputations to ask identification request questions (does not apply to trusted users)
This is if we adopt Option 2 and want to be more strict about the qualities of identification request questions. We can also add one more restriction that they can not ask a second identification request question unless they accept or leave a comment on their first one, given that it has an answer with more than 2 upvotes. This is going a bit extreme, so it should be reserved for the time when the number of low quality identification request questions skyrocketed.
Option 3: Downvote and ignore more the bad questions
Unlocked, unanswered questions that have negative scores will be automatically deleted by the system after 30 days, and questions with -4 score or lower won't show up on the front pagewon't show up on the front page. Plus, downvoting a question doesn't cost one reputationsdoesn't cost one reputations, unlike downvoting an answer which does. Voting is easier and less time consuming than flagging and also saves the reviewers and moderators some time from unnecessary hassle.