Can a (vanilla) Kalman Filter's Observation Matrix $H_k$ depend on the state vector $x_k$?
The non-linear system model that the unscented and extended Kalman filters work with (omitting control) is the following:
$$x_{k+1} = F(k, x_k, v_k) \\ y_k = H(k, x_k, n_k)$$
where $F$ is the state transition function and $H$ is the observation function.  The linear Kalman filter is just the special case where $F$ and $H$ have the form:
$$ F(k, x, v) = A_k x + v \\ H(k, x, n) = B_k x + n$$ for sequences of matrices $A_k$ and $B_k$.  In particular, the functions $F$ and $H$ are linear.

The following is a bit of a pedantic distinction (particularly in the non-linear case) but while the expression $F(k, x_k, v_k)$ clearly depends on $x_k$, the function $F$ itself does not.  In the non-linear case this is no limitation.  In the linear case, if the matrix $B_k$ depended $x_k$ then the function $H$ would no longer be linear.