Relativistic Quantum Mechanics arises as a sort of limit over the degrees of freedom of non-relativistic Quantum Mechanics. Without getting into technicalities, Quantum Mechanics deals with system with finitely many degrees of freedom (1D/2D/3D QHO, a system of finitely many QHOs, etc...). When you take the limit over the degrees of freedom you get into troubles, the main one being that von Neumann's uniqueness does not hold anymore, and one has to deal with a lot of inequivalent irreducible representations of the Weyl algebra (one can even exhibit a family of such representations that has the power of the continuum). To somehow fix this, one considers representations that are reminiscent of the theory with finitely many degrees of freedom: only those representations of the Heisenberg group for infinitely many degrees of freedom that integrates into Weyl form when restricted to any finite subset of degrees of freedom are considered as physically relevant). With this construction one naturally lands into the Fock representation.
In this sense, QM and RQM share a lot in common, the latter being based extensively on the former, but as you can see one has to require something more (like the postulate on the representations of the Heisenberg group). Furthermore, in a relativistic theory you also want to look at it as a dynamical system, where you have an action of the Poincaré group on the algebra of observables. This aspect introduces some more requirements (covariance, spectral condition, locality, ...) that are not considered in the non-relativistic case.