Given the definition of statistical model identifiability : identifiable iff $P_\theta = P_{\theta'} \implies \theta = \theta' ~~ \forall \theta, \theta' \in \Theta$ we can see that, for example, $X \sim N(\frac{\theta_1}{\theta_2},1)$ would not be identifiable.
But can you make a simple, intuitive real-world example of a case when one would meet with an unidentifiable model ?
Another classic example of non-identification is a dose response model like the Emax model, which assumes that some (continuous) outcome for patients $i=1,\ldots,N$ for a drug that you can give in different doses obeys the following: $$Y_i \sim N( E_0 + E_\text{max} \frac{\text{dose}_i}{\text{dose}_i + \text{ED}_{50}}, \sigma^2),$$ where $\text{dose}_i$ is the dose given to patient $i$.
Let's assume we have observed the following data:
| i (patient) | $\text{dose}_i$ | $Y_i$ |
|---|---|---|
| 1 | 0 (placebo) | 1.2 |
| 2 | 0 (placebo) | 0.2 |
| 3 | 0 (placebo) | -0.1 |
| 4 | 0 (placebo) | 0.7 |
| 5 | 0 (placebo) | -0.3 |
| 6 | 0 (placebo) | -0.8 |
| 7 | 100 mg | 11.7 |
| 8 | 100 mg | 12.5 |
| 9 | 100 mg | 13.1 |
| 10 | 100 mg | 11.4 |
| 11 | 100 mg | 10.9 |
| 12 | 100 mg | 11.8 |
The parameters $E_0$ and $\sigma$ are identified by the model/data, but $E_\text{max}$ and $\text{ED}_{50}$ are not. No matter how much data you collect for dose 0 and 100, you will have a whole curve of combinations of those latter two parameters that will fit the data equally well. The way out of this situation is of course to try additional doses of the drug.
Non-identification is not just a consequence of having insufficient observations, and indeed, some of the most pernicious examples of not being able to identify parameters from data arise even if we had access to the entire population, as the below example demonstrates. The discussion below is fairly informal and tries to get at intuition. If you would like to see how statisticians formalize the basic intuition, you may consider reading about the potential outcomes model.
An important example where identification issues play a central role is when one wants to give some causal interpretation to the data. Let us consider the following toy example. We want to find out whether or not the drug Snake Oil decreases the risk of heart attacks. We go out and conduct a survey, and are able to learn (for a moment, suppose that our sample is large enough to avoid inference issues, the following:
In this setting, a $\theta$ we likely care about is the average degree by which Snake Oil causes an increase/decrease the risk of heart attack (i.e. the degree to which making someone take Snake Oil who otherwise would not have taken Snake Oil on average changes their probability in the subsequent year of getting a heart attack). As we will see, the old statistical adage that "correlation does not imply causation" will imply that thes $\theta$ is not identified with the above evidence.
To see why, we show that different values of $\theta$ could be plausibly consistent with the above data. One interpretation of the above data (and one that Snake Oil salespeople will tend to gravitate towards) is that if the population who happened to take Snake Oil had not, they would have been identical to the population who did not take Snake Oil, in which case, the $-1\%$ difference can be fully attributed to the causal effect of Snake Oil on heart attack risk: $\theta = -1\%$. However, the following model might also be consistent with the data: Snake Oil has no effect, $\theta=0\%$, but the population who consumes Snake Oil is especially health conscious. They thus take other actions (some of which are actually effective) to try to improve their health. In this case, the consumption of Snake Oil does not actually cause any differences in heart health, rather, knowing whether or not someone consumed Snake Oil tells us something about what type of person a given individual in the survey is, and these underlying differences in types of people is what leads to the difference in heart health outcomes in the subsequent year.
In quantum mechanics appears many non-identifiable models in the problem of phase estimation. For example, when you try to estimate a phase $\theta \in [0, 2\pi)$ of a qubit, the $X \sim Ber(p(\theta))$, where $p(\theta) = \frac{1}{2}\left[1 + \sin(\theta) \right]$ which is $\pi$-periodic.
