How realistic is the i.i.d assumption in the definition of Shannon's entropy?
No.  Shannon's definition is perfectly general.

There is a special case when the symbols are iid random variables, and you might have seen a formula for that special case (which is indeed simpler), but the definition is fully general.  Note that when we write the entropy $H(X)$, you should take the random variable $X$ to be the entire sequence of symbols.  Then the standard definition applies directly, and doesn't assume that each individual symbol in $X$ is iid.