Experimentally, the uniqueness of the Planck's constant is established through measurements based on the photoelectric effect, Hall effect, black body radiation etc. I don't know of any serious work contradicting this assertion.
However, the operator algebra given in the question with two different "Planck's" constants describes a valid quantum system, although it cannot be obtained as a quantization of the usual commutative algebra of translation operators in the canonical phase space $(\mathbb{R}^{2n}, dp_i\wedge dx^i )$ (at least not in any conventional theory of quantization).
Put in another way, not both $\hbar$s come into existence as a consequence of the quantization process, and the classical theory obtained by the conventional classical limit will depend on their ratio. Thus an independent limiting procedure of the two $\hbar$s like the one adopted in 9503023 mentioned in the comments above, will lead not only to different quantum systems but also to different classical systems.
A constructive way to obtain this operator algebra is to start from a phase space with a noncanonical symplectic structure:
$$ \omega = \frac{\hbar}{\hbar_1}dp_1 \wedge dx^1 + \frac{\hbar}{\hbar_2}dp_2 \wedge dx^2 + ...$$
(This is a special case of a symplectic vector space $\omega$)
In this case, the Poisson brackets will take the form:
$$\left \{x_1, p_1 \right \} = \frac{\hbar_1}{\hbar}$$
$$\left \{x_2, p_2 \right \} = \frac{\hbar_2}{\hbar}$$
When this algebra is quantized according to the rule:
$$\left \{A, B \right \} = C \rightarrow \left [\hat{A} ,\hat{B} \right ] =i \hbar \hat{C}$$
A remark
It is possible to scale the positions and momenta to make the algebra canonical. However mathematically the scaling transformation modifies the symplectic structure, therefore it is not a symplectomorphism; and strictly speaking it describes a different mechanical system.
The uniqueness of the Planck’s constant can be heuristically understood from the path integral quantization point of view. There, the paths are weighted by the complex factor:
$$ e^{ i \frac{S}{\hbar}}$$
Where $S$ is the action. If we believe that there is an action describing all phenomena in nature (theory of everything) and a procedure making the Feynman path integral rigorous, then all systems of nature will be subject to the same Planck’s constant, the one in the denominator of the complex factor.
However, people use different notions of $\hbar$ in different branches of research, please see the articles in nlab about Planck’s constant and deformation quantization.
Let me please go into details of two cases:
In the case of the classical phase space $(\mathbb{R}^{2n}, dp_i \wedge dx^i)$, the algebra of the Hamiltonian vector fields on $ \mathbb{R}^{2n}$ representing the translations on the phase space is commutative. This algebra acts on the functions on the phase space which consist of the classical observables. After quantization, the translation symmetries of the classical system are lifted to act on sections of a line bundle (consisting of the quantum Hilbert space). The lifted algebra is no longer commutative. It receives a central extension. In a given representation of this algebra, the value of the center must be a scalar, since it commutes will all other observables. The value of this scalar is $\hbar$, please see Tuynmann and Wiegerinck). This is also the reason why the two $\hbar$s in the question cannot be of quantum origin, as the geometric quantization process produces a single central extension.
In the problem of quantization of spin can be obtained from the geometric quantization of the two dimensional sphere $S^2$ (please see section 3.5 in the following lecture notes for a concise introduction). In this case, the classical algebra of Hamiltonian vector field is $SU(2)$ and it does not receive a central extension upon quantization. The same algebra acts on the quantum Hilber space. In this case, the Planck’s constant can only introduced as a scaling factor of the symplectic structure:
$$ \omega = \frac{i}{\hbar}\frac{dzd\bar{z}}{(1+\bar{z} z)^2}$$
Where $z$ is the stereographic projection coordinate of the sphere.
The problem of geometric quantization of the sphere has a solution only if the reciprocal of the Planck’s constant ${1}{\hbar}$ is quantized to $2S$ where $S$ is the spin representation of the quantum Hilbert space. The classical limit corresponding to $\hbar \rightarrow 0$ corresponds to $S \rightarrow \infty$. Thus very large spin representations correspond to the classical limit. There are many works discussing the quantization of Planck’s constant, see for example the following work by Eli Hawkins.
These two examples illustrate the difference in the notions of Planck’s constant in different quantum systems and why the conceptual problem of the uniqueness of the Planck’s constant is still open.
