Let's take the simple linear model and discuss three different definitions of the parameter of interest in the linear model, with very different identification settings. All three scenarios are very common in empirical work in economics and it's typically only clear from context of them characterise the analysis.
Here's the linear model.
$$y_i = x_i^T \beta + \epsilon_i$$
It's clear what $y_i$ and $x_i$ are. But what are $\beta$ and $\epsilon$ supposed to be? The answer determines how we should think about identification.
1.) Prediction
We could simply be interested in predicting the observable $y_i$ when we see the observable $x_i$, using the best linear function of $x_i$ to do so. Then the target parameter should be defined as $\beta := \arg \min_{b \in R^k} E[ (y_i - x_i^T b)^2]$ and the error as $\epsilon_i := y_i - x_i^T\beta$. The expectation $E[\cdot]$ is with respect to the distribution of observables. This distribution is identified if we observe $y_i$ and $x_i$. The solution for $\beta$ is the well known formula $E[x_ix_i^T]^{-1} E[x_iy_i]$. Note that $E[\epsilon_i x_i^T]=0$ holds by construction. If we find the best linear predictor, then the prediction error doesn't have a linear relationship with the predictor. If it did, we would have failed at finding the best linear predictor! What about $E[\epsilon_i |x_i]$? That will be zero only if $E[y_i|x_i] = x_i ^T \beta$, in which case $x_i ^T \beta$ isn't just the best linear predictor but the best predictor. The only failure of identification for this definition of $\beta$ would be that $E[x_ix_i^T]^{-1}$ might not exist, perhaps because some of the $x_i$ are linear functions of others. Then $\beta$ is not identified, there are different $\beta$ that all do the job of forming the best linear predictor. But that wouldn't worry us because we just want to predict and they are all equally good at that.
2.) Prediction with the underlying, not the measured covariate
Often, $x_i$ is a proxy for something we are interested in. If $x_i$ is the body mass index we might be interested in how it predicts an outcome but we only get to work with a self reported estimate of it, $w_i$. If we are content with finding out the predictive relationship between that and an outcome we are back in the first scenario. If not, we face an identification problem. We are interested in $\beta$ as defined in the previous scenario, but we do not observe $x_i$. We observe only $w_i = x_i + u_i$. Now $\beta$ is not identified. We have $y_i = (w_i - u_it)^T \beta+ \epsilon_i$ or $y_i = w_i^T \beta+ \epsilon_i - u_i \beta$. And now $E[(\epsilon_i - u_i \beta) w_i^T]$ is no longer $0$ by construction and OLS won't estimate $\beta$ The most common identification strategy for this problem is to find an instrumental variable: Something that correlates with $x_i$, but not with $u_i$, and not with $\epsilon_i$ either. Maybe a friend's anonymous estimate of that person's body mass index. Note how nothing so far has been about causality, just about prediction, but there is an identification problem anyway, because an important variable is unobserved.
3.) Causal inference
Probably the most common scenario: We actually want to know how $x_i$ affects $y_i$ not just how it predicts it. How should we define that? We can use structural equations to do it, which express how a variable really, is determined: $y = f(x, \epsilon)$. This unknown function tells us what the value $y$ would be if we set $x$ for any possible value, just like proper scientific models do. It also depends on $\epsilon$ which is now the effect of other causes of $y$ and this may well be correlated with $x$ in the data we see. Let's suppose $f$ is linear: $y_i = x_i^T \beta + \epsilon_i$ as before. This $\beta$ is the one from the first scenario only if $\epsilon_i$ is uncorrelated with $x_i$. In most empirical analyses, one can think of reasons why $x_i$ and $\epsilon_i$ might be correlated, and so an identification strategy is needed.
To clearly distinguish this scenario from the first one, different notation is sometimes used: The potential outcomes notation, or causal graphs, or structural models. The best way of thinking about causality in economics is an ongoing area of research and there is no consensus yet. As it stands most researchers will will simply use the standard regression equation, expected values etc and you need to infer from the context whether the goal is prediction, prediction with an imperfectly measured covariate, or causal inference.
A very good textbook covering this material is Mostly Harmless Econometrics.
