Why are there a bunch of anti-something fan clubs? I thought fan clubs
were for fans of a character, not for people who dislike a character. Isn't
this contradictory?

I think a more proper term for these types of "fan clubs" is:
[Character] Hate Club

Example:
Instead of "Anti-Cynder Fan Club," have a "Cynder Hate Club."

You have a point

I think a more proper term for these types of "fan clubs" is:
[Character] Hate Club

Example:
Instead of "Anti-Cynder Fan Club," have a "Cynder Hate Club."