. The condition restricts the available Hilbert spaces to 
, 
, 
, 
, for finite dimensional quantum systems, and 
for infinite dimensional ones. Clearly the available Hilbert spaces are countable. If one takes the position that systems with the same dimensions are equivalent, then we arrive at the conclusion that there are countable number of quantum systems.
I am aware of at least one credible instance where equality of dimensions was taken to imply the equivalence of quantum systems. Arno Bohm, in his “The Rigged Hilbert Space and Quantum Mechanics”, motivated the introduction of the rigged Hilbert space as a replacement of the separable Hilbert space of standard quantum mechanics with the premise “that all (separable infinite dimensional) Hilbert spaces are isomorphic and so are—roughly speaking—their algebras of operators. This means that all physical systems would be equivalent, which is obviously not the case (unless one has only one physical system, which then would have to be the microphysical world).” This claim, pushed to its logical conclusion, implies that finite dimensional quantum systems with equal dimensions are as well equivalent because a finite dimensional Hilbert space is unique up to a unitary transformation.
But what is a quantum system? A quantum system is ultimately defined by the measurements and the outcomes of measurements performed on the system. Consider a photon and an electron. If we restrict our attention to the internal degrees of freedom of the photon and the electron, then we will be restricted in their polarization and spin degrees of freedom, respectively. The photon has vertical and horizontal polarizations; these polarization states are mutually exclusive and hence define a basis for the internal degree of freedom of the photon. The Hilbert space is two dimensional and is given by . On the other hand, the electron has spin-up and spin-down states; these spin states are also mutually exclusive and define a basis for the spin degree of freedom of the electron. The Hilbert space is again .
The two internal degrees of freedom reside in two isomorphically equivalent Hilbert spaces, yet they are clearly different physical systems. They are different systems because they involve two different sets of measuring instruments in identifying them. The photon polarization is identified with measurements involving, say, polarizers and beam splitters; on the other hand, the electron spin is identified with measurements involving, say, the magnetic field interacting with the magnetic moment of the electron. These two sets of measurements are not interchangeable. The electron spin cannot be measured by passing it through a polarizer, and the photon polarization cannot be measured by magnetic interaction.
While it is true that a quantum system is a Hilbert space in its barest mathematical sense, a separable Hilbert space is not a single quantum system but an abstraction of a family of quantum systems characterized by their common dimensions. A specific quantum system is a particular physical realization of a Hilbert space, but it is by no means the only realization. There are potentially infinitely many possible realizations of a given Hilbert space. One only has to recognize the infinite class of interaction potentials that may act on a quantum particle—all representing distinct physical systems. So while the available separable Hilbert spaces are countable, the available quantum systems are uncountably many.