If your system is ergodic (meaning it randomly occupies all possible microstates over long time periods) and has a discrete number of microstates, you could measure entropy by repeatedly measuring which microstate it's in and computing $S = \sum_i p_i \log(p_i)$. This is horribly impractical, because it requires knowledge of and access to a system's microstates, and enough time to build up a good guess for the probability distribution $p_i$. But at least it's in principle a passive operation; we don't have to dramatically change the system to get the quantity that we want.
Without the above expression, we're forced to course-grain over microstates and take the perspective of thermodynamics. All other conventional relations that involve entropy, like $dS=\delta Q_\text{rev}/T$, are ultimately expressions for change in entropy, so you can't use them to compute absolute entropy without doing drastic things to your system. To measure entropy using relations like $dS=(dU+PdV)/T$, we have to compute changes in entropy until we get the system to a state of known entropy, like absolute zero (which is usually zero but may not be for some tricky systems like glasses). But this is far from a passive operation! So while this means that there is an experiment you can do to measure entropy, it will be far more invasive than using a thermometer.
So why isn't there a simple thermodynamic expression for total entropy? The following isn't a proof, but I think it captures the essential reason. The operational, and therefore easily measurable thermodynamic quantities like temperature and chemical potential are derived by maximizing entropy, and to do that we take the derivative of entropy and set it to zero. For example, to derive temperature as a concept, we consider two systems that are allowed to exchange energy and compute how total entropy changes as they do so
$$\frac{\partial S_{tot}}{\partial E_{1\rightarrow 2}} = \frac{\partial S_1}{\partial E_{1\rightarrow 2}} + \frac{\partial S_2}{E_{1\rightarrow 2}} = -\frac{\partial S_1}{\partial E_{2\rightarrow 1}} + \frac{\partial S_2}{E_{1\rightarrow 2}}$$
The second equality follows because energy flowing into system 2 has to equal negative the energy flowing into system 1. In equilibrium, $S_{tot}$ is maximized and $\partial S_{tot}/\partial E_{1\rightarrow 2} = 0$, so we get $\partial S_1/\partial E_1 = \partial S_2/\partial E_2$, which we define to be $1/T$. The concept of entropy leads to the operational quantities like temperature when we compute how entropy changes. Therefore the resulting quantities are insensitive to the absolute magnitude of $S$.
