Is (energetic) degeneracy a physical property?

In quantum mechanics, observable properties correspond to expectation- or eigenvalues of (hermitian) operators.

After measurement (of an eigenvalue) the system is in an eigenstate that corresponds to an eigenfunction of the operator. Sometimes, however an eigenvalue ($lambda$) can not only arise from one certain well-defined state-function but from a whole (vector) space that is spanned by $n$ eigenfunctions with the same eigenvalue $lambda$ for $n>1$.

What I am interested in, is if the degeneracy itself is something that can be measured? For example can there exist an operator, say $mathcal{D}_mathcal{H}$ that measures the degeneracy of a state function $Psi_lambda$ (for a certain operator, for example the Hamiltonian $mathcal{H})$:

$$ mathcal{D}_mathcal{H} Psi_lambda = n Psi_lambda $$

This question arises from some considerations of special symmetry properties of degenerate states I am investigating. In the course of these works the question arose, if the degeneracy of a state is some physical property or rather only something like an "mathematical artifact" that cannot be directly probed experimentally.

Note on a side: I can imagine that the case is for "symmetry imposed degeneracy" where the state corresponds to higher-dimensional irreducible representations is different than for the general thing which shall include what is sometimes called "random degeneracy".