A realistic example of a non identifiable model? 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 ?

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:

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*Among those taking Snake Oil in a given year, 1% of individuals have a heart attack in the subsequent year

*Among those not taking Snake Oil in a given year, 2% of individuals have a heart attack in the subsequent year

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.
