Why is the expected value (mean) of a variable written using square brackets? My question is told in a few words: Why do you write $E[X]$ in square brackets instead of something like $E(X)$? Probably it is not a "function". How would you call it then? This question also applies for $Var[X]$.

I don't. 
You can write in both ways, it doesn't matter. Some don't even use brackets but might instead write $EX$ or $VX$. 
