Wikipedia is a community-built site, so the closest thing you can get to "official" are the policy pages, which are adopted based on community consensus. But even then, enforcement can be inconsistent and spotty, especially with the more obscure articles (which is why editors often dismiss the "Other stuff exists" argument as invalid).
So on a Wikipedia article, there are two main places where the anime/manga's genre can be specified:
1. The animanga infobox (at the top).
The Infobox animanga template has these guidelines for the genre field:
List more specific genres over general genres (ex. Mecha is a subgenre of Science fiction, therefore only Mecha should be listed instead of both). Genres should be based on what reliable sources list them and not on personal interpretations. If a genre cannot be cited to a reliable source, then it should not be listed.
The genres in the infobox depends a lot on what reliable sources characterize the anime/manga to be. Even if it seems very obvious to the common reader that the anime/manga is an instance of a specific genre, including it in the infobox is no good unless you can find a reliable source supporting it. (You may find some Wikipedia articles that have genres in the infobox that are unsourced, but shhhh, that's just the inconsistent enforcement thing from earlier.)
As a result, there isn't really a well-structured organizational hierarchy for the genres in the infobox. It's more like a mish-mash of what is considered to be a genre from various sources.
2. The article's categories (at the bottom).
Per Wikipedia 's Manual of Style for anime/manga-related articles, anime/manga articles are generally put into one or more subcategories of the three parent categories:
Editors are generally more lax with respect to verification requirements for categories (a category generally sticks around as long as it's at least somewhat apparent from the article content), but categories do get removed or challenged (e.g. using Template:Uncited category).
Again, Wikipedia is a community-built site, so there isn't necessarily a consistent scheme or structure to the genre hierarchy. The structure of the genre categories most likely took shape over numerous small changes by various passerby contributors. Additions or changes to categories are usually implemented boldly. Though larger or possibly controversial changes are typically posted at Wikipedia:Categories for discussion first to solicit feedback or approval.
This noticeboard isn't as popular as the Articles for Deletion noticeboard by a longshot, but things typically get one or two responses here. As an example of the kind of discussions that can take place here, here's a fairly straightforward bulk category renaming proposal from 2018, which someone proposed so that the subcategories could have a more consistent naming scheme.
