For simplicity, let's consider $n=2$. In $U$, we have $|f|+|g|=C$ for some constant. Fix a point $z_0$ in $U$. Then, for appropriately chosen unimodular constants
$$|\alpha f(z_0) +\beta g(z_0)|=|f(z_0)|+|g(z_0)|=C.$$
This means the holomorphic function $\alpha f(z) +\beta g(z)$ attains its maximum in $U$ (since the supremum in $U$ is at most $C$ by the triangle inequality, and it attains $C$), so it is a constant. So for all $z$, we have
$$|\alpha f(z) +\beta g(z)|=|f(z)|+|g(z)|.$$
Equality is only possible in the triangle inequality all the vectors point in the same direction. So $c(z)\alpha f(z) = \beta g(z)$ for some real holomorphic $c(z)$. But since real holomorphic functions are constant, $c$ is constant. Then $\alpha f(z) +\beta g(z)=(1+c)(\alpha f)$, and the latter is a holomorphic function that attains its maximum on $U$, so it is constant. So $f$ is constant, and it follows that $g$ is constant.