On the proof of that the only linear operator mapping each element of a complex vector space to an orthogonal element is the $0$ operator
It is the polarization identity for the bilinear form $$ B(u,w)=\langle Tu,w\rangle. $$ The identity that you mention is a particular case when $T=Id$. In general, the identity relates the bilinear form to the corresponding quadratic form.
The polarization identity is invoked whenever you want to conclude something about the dot product (angles, etc.) based on information about lengths. For example, you can prove that a linear transformation preserving lengths also preserves angles, areas, etc.