The analogy to Galilean relativity actually is too kind. Galilean relativity relies on Euclidean space as a background, allowing an external viewpoint fixed to empty coordinates. Hilbert space is not a background space at all; it is always defined in terms of physical systems, what is known as a constructed space. You can transform perspectives in spacetime, but there is no transformation to a background perspective in Hilbert space because no such background exists. The closest that exists is a statistical transformation to different perspectives within Liouville space, but this only works for objects within the space; you cannot transform to the perspective of the background itself as it is not a background space.

...which is why eventually you need to switch to the grown-up version of Quantum Mechanics, Quantum Field Theory, is defined in terms of relativistic fields with a single "universal" field for each flavor of particle.