There is no necessary reference to any particular physical system when thinking abstractly about this system and its observables, yet once we have, we can use it to describe any two-level physical system (I think), e.g. a coin that can be heads or tails. Is this the case in quantum mechanics?
Yes, you can think abstractly about a system with no necessary reference to any particular physical system. But it is not necessarily a helpful way to proceed. The two state system is a model system that might or might not provide a useful approximate model for a real physical system.
As illustrated below, you can proceed in complete generality with the two state system and its observables, if you would like.
Let's say we use the orthonormal $\{|H\rangle, |T\rangle\}$ basis where $\sigma_3 |H\rangle = |H\rangle$ and $\sigma_3 |T\rangle = -|T\rangle$. For a two-state system, the Hermitian operators that can represent observables are real linear combinations of the Pauli operators $\sigma_1$, $\sigma_2$, $\sigma_3$ and the identity operator $I_2$. However, unlike the classical case, it seems that when applying this abstraction to a physical system, there are not always candidate observables for all of them apart from $a \sigma_3+ b I_2$, i.e. the observables that are diagonal for the choice of basis.
In general an observable is a Hermitian operator. (The converse is not true, as we discuss further in the update section below.) The eigenvalues of the observable represent the possible outcomes of a measurement of the observable.
For your two state systems, the most general Hermitian operator can be represented in matrix form as:
$$
M = \begin{bmatrix}a & b+ic\\b-ic & d\end{bmatrix}\;,
$$
where $a$, $b$, $c$, and $d$ are real.
The eigenvalues of $M$ are:
$$
\lambda_M = \frac{a+d}{2}\pm\frac{\sqrt{(a-d)^2+4|b+ic|^2}}{2}
$$
The most general state has the form:
$$
|\psi> = \alpha|H> + \beta|T>\;,
$$
where $\alpha$ and $\beta$ are complex numbers. And we also impose the normalization condition
$$|\alpha|^2 + |\beta|^2 = 1
$$
The expected value of the above-mentioned general Hermitian operator in the above-mentioned general state is:
$$
<\psi|M|\psi> = a|\alpha|^2 + 2\Re(\alpha^*\beta(b+ic))+ d|\beta|^2\;,
$$
subject to the normalization condition.
This is all completely general, and in my opinion, all pretty opaque.
But. it is still useful to have these formulas since you can apply them to any specific cases as well. E.g., if the you are interested in a Hamiltonian of a coupled two level system like:
$$
H = \begin{bmatrix}-\epsilon/2 & v\\ v & \epsilon/2\end{bmatrix}\;,
$$
where the lower level is at energy $-\epsilon/2$, the upper level at energy $+\epsilon/2$ and the coupling is a real number $v$. And you can read off the observable energy levels immediately from the above formula as:
$$
E_{\pm} = \pm \frac{\sqrt{\epsilon^2+4v^2}}{2}\;.
$$
In practice you might, say, tune these parameters to model some energy splitting from an experiment.
Update:
I didn't notice there was a question in the title... which reads:
"For a generic two-state quantum system, are there interpretations for the observables corresponding to all Hermitian operators?" [Emphasis added.]
We note that "are there interpretations for..." is really asking. Are there interpretations for the following equation:
$$
\lambda_M = \frac{a+d}{2}\pm\frac{\sqrt{(a-d)^2+4|b+ic|^2}}{2}
$$
Or, rather, can we come up with some "other" meaning beside that of a pure function for the following function of four real variables:
$$
\lambda_M(a, b, c, d) = \frac{a+d}{2}\pm\frac{\sqrt{(a-d)^2+4|b+ic|^2}}{2}
$$
We provided one possible meaning for
$$
\lambda_M(\epsilon/2, v, 0, -\epsilon/2)
$$
as energy levels of a coupled two-state system.
There are clearly other interpretations. This is up to the imagination of the reader.
Now, to the word "observable" and its relation to a Hermitian operator. Any Hermitian operator whose eigenvectors span the space (in this case a two-dimensional space) is an observable. (See, for example, Messiah's Quantum Mechanics textbook, volume 1, chapter VII, second 9 titled "Eigenvalue problems and Observables.")
Finally, the issue of super-selection rules, where the Hilbert space is a direct sum of spaces, is not very relevant in our present analysis (the analysis of a two-state system). This is because we have already agreed that the one-state system is trivial; the only possible direct-sum to consider would be a direct sum of trivial systems.
