I would argue that we should not allow these types of questions. Here is what is listed as on-topic:
- Plot, character, or setting explanations
- Historical or societal context of an anime or manga
- Anime and manga identification
- Franchise/series reading or viewing order
- Anime and manga production
- Tropes, terminology and other general anime and manga related information.
The closest that this gets would be to being anime and manga related information, with a bit of identification request as well, but it doesn't have the answerability of either of them.
The problem with asking about AMVs is that there can be dozens of AMVs that fit the same characteristics, and AMVs can also be taken down from Youtube or wherever else they're posted.
For the AMVs that have been asked about, one has been identified so far. It describes a scene, the song, and the physical event that they saw it at. For the others, one names the song, the anime, and a speech made during it and the other names the anime and describes the lyrics of the song.
It would be possible to put similar restrictions to what we currently have for anime/manga identification questions, but that wouldn't necessarily help, because AMVs tend to be made in packs (many people make AMVs for the same anime using the same song).
For example, let's say someone had this description:
I'm looking for an AMV I saw about 5 months ago with the song "In the End" by Linkin Park. I know it focused on the fight between Naruto and Pain in the anime Naruto Shippuden.
That description gives the year range, the song (with the artist), the anime, and the specific part of the anime that was focused on. That is 4 points of description, which in the regular anime/manga identification request would be enough to to make it okay. In this case, though, there are at least a dozen AMVs that fit that criteria. There will be no way to require enough criteria that it could reliably narrow it down, especially for the more popular anime. Additionally, there is no way to know what AMVs have been removed from Youtube or other websites. Especially if someone is asking for an old AMV, it might just not exist anymore.
There is also no way to know if the person was right. Let's say there are three AMVs currently on Youtube that fit the description, but the OP doesn't remember any of them. That doesn't necessarily mean that the answer is wrong--it just means that they can't remember.
The only type of AMV request that I could see as being reliably answerable is one like the one that has been answered, where it was shown at an anime convention, so there would be some reliable record of what it was. That also relates to anime conventions, which is maybe(?) on-topic.
Tl;dr: No because they are too broad and not necessarily answerable.