Just because you see plenty of those doss not mean they are doing it legally (and many of them can have their revenue diverted to the copyright holder)

There are specifics for Fair Use and typically fan cover songs does not fall under it. These include:
Transformative works
Educational purposes
Commentary purposes (e.g. used for discussion)
A parody work (not satirical, see the link at the end of the comment)

Many of the AMVs you see often commits two copyright violations: the unlicensed use of the anime and unlicensed use of the music.

Why do they remain up instead of being taken down? Several reasons:
The company(ies) did not notice them.
The copyright owner allowed the video to remain up but subsumed their ad revenue
Really rare instance but they may be a fan of the video.

What about those doing the covers like AmaLee?
Usually they have a license for sampling of the instrumental music
If the lyrics are from another language, typically the translated lyrics for many of these covers are not 1:1 translations and have enough artistic differences that it could be considered non-infringing (depending on the content per song).

But when is Fair Use actually considered? Typically when you are sued - the court will determine if what you did was actually under Fair Use or not.

Related links: