"Identification" is the most loaded term in econometrics. There are multiple cheap talk equilibria with regard to its meaning. It is used with different intended (but related and overlapping) meanings, in different contexts, by people with different orientations, with different levels of precision.
Therefore you will get a range of correct answers.
Here is an attempt covering a few of the variations, going from the theoretical end of the spectrum to the empirical.
Statistics
A statistical model is a one-to-one mapping $\theta \mapsto P_{\theta}$ from a given parameter space to a family of probability measures.
It is the one-to-one property of the mapping that makes the model "identified".
No two different elements in the parameter space can give rise to observationally equivalent data generating processes.
In statistics, a model is therefore, by definition/assumption, always identified.
(This can be seen in the assumptions for all the foundational results, e.g. Neyman-Pearson.) Statisticians never speak about identification, because they don't have to.
For example, for
$$
y = \beta x + \epsilon \quad (*)
$$
where $(x,\epsilon)$ is bivariate normal, to specify a model for population $(x,y)$ parametrized by $\beta$, one must assume that $Cov(x, \epsilon) = 0$. Without imposing this assumption, different $\beta$'s could give rise to the same distribution for $(x,y)$. In econometrics, which is much more explicit about the identification issue, the condition $Cov(x, \epsilon) = 0$ will sometimes be called an identification assumption.
Structural Models
If one tries to construct a statistical model by adding unobserved disturbances to an economic model, identification needs to be addressed.
In order to make the resulting structural econometric model identified, usually one needs to make certain assumptions, either of economic or technical nature. These are called identification assumptions.
For example, suppose there are $n$ firms in Cournot competition with private constant marginal costs $(c_1, \cdots, c_n)$ drawn from joint density $f(x_1, \cdots, x_n)$.
The econometrician observes firms' output $(q_1, \cdots, q_n)$ and market price $P$ and would like to identify $f$. One possible identification assumption is that the Jacobian of the FOC system
$$
\frac{d P(Q)}{dQ} q_i + P(Q) - c_i = 0, \, i = 1, \cdots, n,\, \mbox{ where } Q=\sum_1^n q_i
$$
is non-vanishing. Then, by the Implicit Function Theorem, $(q_1, \cdots, q_n)$ maps one-to-one locally to $(c_1, \cdots, c_n)$. This implies the model, parametrized by the observed quantity $(q_1, \cdots, q_n)$, is identified, at least locally.
The empirical interpretation is that sufficient variation in the trade-offs faced by the firms
allows you to identify $f$.
There are more interesting examples where the identification assumption
puts restriction on economic agent behavior, etc.
Empirical Usage-Consistent Estimation
So far, identification is purely a property of the mapping from parameter to data generating processes. Identification is a prerequisite for estimation but by itself it makes no mention of the sample.
There are also contexts where an econometrician speaks about a specific estimator that's designed to estimate a specific parameter in a specific model.
An assumption under which the estimator consistently estimates the parameter is called an identification assumption.
For example, given time-series data $(x_t, y_t)$ generated by
$$
y_t = \beta x_t + \epsilon_t, \; t = 1, 2, \cdots, \quad (**)
$$
the parameter $\beta$ "can be identified by OLS $\hat{\beta}$" under the assumption that $Cov(x, \epsilon) = 0$.
In $(*)$ and $(**)$, the condition $Cov(x, \epsilon) = 0$ and the terminology are the same, but "identification assumption" have different (but clearly related) meanings.
Empirical Usage-Causal Inference
When one is interested in establishing causal effect, a condition imposed on the model that allows for causal interpretation of the estimate is called an identification assumption. Yes---$Cov(x, \epsilon) = 0$ for the linear model would fall under this category also. Often it is strengthened to $E[\epsilon|x] = 0$, which is more interpretable for causal inference.
Similarly, when $Z$ is an instrument, the exogeneity condition $Cov(Z, \epsilon) = 0$ is an identification assumption.
For diff-in-diff, the parallel-trends condition is an identification assumption.
For regression discontinuity design, the identification assumptions are that, first, there are no other discontinuities except the forcing variable, and second, agents cannot manipulate the forcing variable.
The corresponding empirical design (e.g. IV/DID/RDD/etc) is sometimes called the identification strategy.
In this context, "identification" is not a binary condition. One could have weak identification, e.g. a weak instrument.
Used in this sense, an identification assumption clearly needs to be justified when it's claimed to hold empirically. In other words, one needs to justify that the corresponding variation is exogenous---e.g. the variation of the instrument is exogenous, etc.
In your quoted example,
The paper examines the effect of bank runs on lending. We exploit
variation in the structure of banks' liabilities to identify banks
that were more vulnerable to the run...
vulnerability to a run is clearly a endogenous variable in relation to lending. The claim is then the empirical design in question uses exogenous variation in the structure of banks' liabilities---as an instrument/forcing variable/whatever---to circumvent endogeneity and achieve identification.
