Let $D\subset\mathbb{C}$ be the open unit disk. Suppose $f,g,F,G:D\rightarrow\mathbb{C}$ are analytic functions linked by 
$$\vert f(z)\vert^2+\vert g(z)\vert^2=\vert F(z)\vert^2+\vert G(z)\vert^2; \qquad \forall z\in D.$$



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> **Question 1.** If $f\neq \alpha g$ and $g\neq\beta f$ for any $\alpha, \beta\in\mathbb{C}$ then is the same true for $F$ and $G$?
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**Caveat.** Not true for real analytic functions: 
take $f=\sin x, \,g=\cos x, \, F=\frac1{\sqrt{2}}=G$.


**EDIT.** Encouraged by Christian's positive answer, let's upgrade the problem.



> 
> **Question 2.** Suppose $f, g, h$ are linearly independent (over $\mathbb{C}$) such that 
>  $$\vert f\vert^2+\vert g\vert^2+\vert h\vert^2=\vert F\vert^2+\vert G\vert^2+\vert H\vert^2.$$
>  Should $F, G, H$ be linearly independent, too?
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modulus-related analytic functions
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".


