modulus-related analytic functions

Yes, for Question 1.

We can divide through by one of the functions on the left-hand side and also assume $F=G$, and then the question becomes: Can $1+|h(z)|^2$ be the square of the absolute value of a holomorphic function, for a non-constant holomorphic $h$?

For this to happen, we need $\log (1+|h(z)|^2)$ to be harmonic. If you now just work out the Laplacian of this function and use the Cauchy-Riemann equations for $h$, you'll find that the numerator of the expression you obtain from two applications of the quotient rule equals $4((h_1)_x^2+(h_2)_x^2)$, so $h$ has to be constant.

In fact, more is true. If $f_j$ are linearly independent and $F_i$ are linearly independent, and $$\sum_{j=1}^n|f_j|^2=\sum_{i=1}^m|F_i|^2,$$ then $m=n$ and $F_i$ are obtained from $g_i$ by a unitary transformation. See, for example, https://arxiv.org/pdf/math/0007030.pdf, section 3. This is called the "Calabi rigidity".