There might be problems, but I think "overanalyze" might not be the term we're looking for. As far as I can tell, "overanalyze" just means to analyze in too much detail. I think this is a little too vague and subjective to be forming site policy on.
I actually think a lot of my questions and answers would fall under the category of overanalysis, such as this one: Why did Hiruzen not meet with Fugaku? and this one: Did Miki really never bully Shoko?. Or this: Has Miko Ino's morality been corrupted?. But I'm not sure this is what you're actually worried about or not. Based on what I'm reading, I think you might mean something different. Correct me if I'm wrong, but I believe you might want to know whether it's okay to ask questions that seem to be analysis questions on the surface, but are actually complaints about a series or it's characters.
I think this problem needs to be broken down into different parts, which might need additional Meta questions (or might already be answered).
"This Show is Ridiculous" Masquerading as Analysis Questions
Sometimes users ask a question, but what they're really doing is complaining about a show.
From time to time, we get questions like this (this is one of mine): Is there any explanation for how Akihiko Kayaba managed to trap everyone in SAO?
Perhaps a better example of a bad question: Why didn't Ash release Charizard?
These questions can be annoying, but often they can be objectively answered with a little (or a lot) of overexplaining. Though sometimes the answer is so obvious that it's questionable whether anything of value to the site is generated. I'm not sure if we have any Meta questions about these types of questions. If not, perhaps someone should ask. This is probably the closest thing we have right now: How should we deal with posts that complain about the plot/characters/author?
Users Treating the Site Like a Forum
This can happen with the previous category, if one gives an objective answer and the user doesn't accept it, and ends up writing a lot of comments and being chatty. I would say this is actually a user behavior problem, so the solution isn't close votes, but rather to explain the rules to the user. Though if the user is inviting discussion in a question (for example, by asking whether Superman or Goku would win in a fight), that can usually be closed on the grounds of opinion-based.
Reality Check Questions
Such as this one: How is Misaka's railgun so destructive? or the previously mentioned Why was the Otsusuki clan genetically compatible with humans to the point they could have children?. These would seem to already have a Meta post: Is it ever acceptable to ask about whether an anime event is realistic or plausible?
Personally, I kind of like questions like these, and I especially like seeing the clever answers we sometimes get. The problem is whether the users on here have the expertise and patience for answering them. I don't know whether or not they should be on topic.
As far as overanalyzing and analyzing go, I don't think these questions should be closed as being out of the scope of the site unless they're obviously going to generate opinion-based answers. However, I can see overanalyzing questions being annoying, as they might require an excessive amount of work to anyone trying to answer. If someone is willing to do the work, then the question can be answered. If not, it just gets ignored and maybe downvoted.
And we can perhaps take a note from the Literature site, where they say literary analysis is on topic: https://literature.meta.stackexchange.com/questions/42/is-literary-analysis-on-topic