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

>**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 $G$?

**Caveat.** Not true for real analytic functions: 
take $f=\sin x, \,g=\cos x, \, F=\sqrt{\frac12}=G$.


**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?