The whole sentence is probably one I found in a Google search. When you are asking a question like this, try to include as much context as possible (a whole sentence, and maybe the sentence before it), and try to show your source.

"Video game violence isn't real-world violence, any more than chess is an accurate simulation of war, but many people feel the link is there."

The question is whether participating in fantasy violence makes people likely to commit real violence. Some people think that it does. The writer is suggesting that it doesn't.

Chess is a kind of war game. But it is so abstract, so different from real war, that it does not carry the emotional weight of real war. Nobody thinks that playing chess will make people to go out and kill real bishops and queens. Chess is not an accurate simulation of war. Nobody trains officers to fight a real war by having them play chess.

Video game violence isn't like real-world violence.

The writer is saying "Video game violence isn't like real-world violence any more than chess is like war."

How much is chess like war? Let's pretend we could measure it. Let's say chess is a 1%-realistic simulation of war. How much is video-game violence like real world violence?

Some might say "video-game violence is 10%-realistic. It is much more realistic than chess."

The writer is saying "video-game violence is not real-world violence, any more than chess is an accurate simulation of war." Taking the phrase literally, video-game violence is only 1%-realistic or less.

I'm just using numbers as a way to explain the phrase.

"X is not Y, any more than Z is not W" is a way of saying "X is to Y as Z is to W. You understand and agree that Z is not W, so you should also agree that X is not Y."

I think you're missing an important part of the phrasing which goes something like "(the thing is) no more/less xx any more than chess is an accurate simulation."

Ex: I am no more a landscaper because I planted a tree in the backyard any more than my neighbor is an artist because he's painting his house.

It's expressing a "not true" thing (negation/doubt) by comparing it to another "not true" thing.

Hi

Just to add something to Henry's answer which I find very accurate,

It means that you can believe that you are good in war just because you are good at chess!

'Any more than' compares two things, but your quote is missing the first half. Look at this sentence:

In this case, it compares two things: going to church and going to a garage. Just because you go to church you are not a Christian. Just because you are in a garage you are not a car. 'Any more than' can be thought of as 'not more than' or 'just like'. It is really only used to compare two things which are bad.

I'm not a native speaker but it sound to me like a bad English or missing a wider context.