Why is the polynomial in AES MixColumns multiplied modulo a reducible polynomial?
The reason it is not irreducible is because it does not have to be.

For MixColumns, the result must be a polynomial of degree 3 or less, which requires a degree 4 reduction. We are not reducing the elements of the finite field here, but the coefficients of the polynomial generated by multiplication of the fixed and input polynomials. With reduction by $x^4+1$, the reduction can also be represented as replacing the coefficient of $x^0$ with that of $x^4$.

The choice of degree 4 reduction polynomial does not have to be irreducible in $GF(2^8)$ if the fixed polynomial in the matrix multiplication has an inverse, which it does. In the case of $c(x)=3x^3+x^2+x+2$, $c^{-1}(x)=11x^3+13x^2+9x+14$.