Short answer since I'm on mobile:
No, it's not a postulate but rather a theorem. First, clear things up a little. We want a symmetry of the theory to act as an arbitrary transformation which conserves the unitarity of our theory. A transformation which does not act as a symmetry of our system need not to be a linear transformation.
Now, for symmetries, there exists a famous theorem by the mathematican Eugene Wigner, known as Wigner's Theorem, which states that the transformation encoding the action of this symmetry on the Hilbert space of states must be a linear transformation.
So to use your example, merely a change of reference frames is of course a symmetry and therefore by Wigner's theorem acts simply as a linear transformation.