What is special about the theorem that over GF(2), $f^2(X) = f(X^2)$?
You are confusing polynomials, elements of $\text{GF}(2)[X]$, with coefficients, elements of $\text{GF}(2)$.  Admittedly, it's certainly the case that as functions $\text{GF}(2)\to\text{GF}(2)$, any polynomial (or any function at all) will be equal to its (pointwise) power for any exponent greater than 0 which no doubt adds to the confusion.  Still, as polynomials (roughly, finite lists of coefficients), they need not be equal.  For example, $X^2 \neq X$ in $\text{GF}(2)[X]$ simply by definition.  With fields of characteristic $0$, evaluation is injective as a function $F[X]\to F^F$, so conflating polynomials with the functions they represent is not as much of a problem.

So for $f(X) = 1+X$, $$f(X)^3 = (1+X)^3 = 1 + 3X + 3X^2 + X^3 = 1 + X + X^2 + X^3 \neq 1+X^3 = f(X^3)$$