Nested Individuals within a group - meaning? I am reading "DID Intro" by Stata Corp LLC.
It is recently updated, early April 2022.
On page 7, it explains how Stata runs a panel-data difference-in-differences regression.
My question is what do the following two quotes mean?
The first quote is (p. 7):

"Individuals are
assumed to be nested within the group, and treatment occurs at the group and time levels."

The second quote is (p. 7):

"... is fit so that data comprising a panel of individuals is observed across the time
for which individuals are nested within treatment group."

Specifically, I am trying to understand what difference does this  imply in terms of having "repeated cross-section" dataset as opposed to panel dataset for a DiD analysis.
 A: 
"Individuals are assumed to be nested within the group, and treatment occurs at the group and time levels."

In other words, the individuals are embedded within a group. For example, the individuals $i$ may work or reside within state $s$. Here, we often say individuals are "aggregated" by state; they are contained within this higher level contextual setting.
Note that on page 7, the authors introduce the following model,
$$
y_{ist} = \gamma_i + \lambda_t + \delta D_{st} + \textbf{z}_{ist}\beta + \epsilon_{ist},
$$
where we include individual (not state) fixed effects. We can estimate the fixed effects at this lower level because we observe the same individuals within each state before and after the treatment. We cannot do this with repeated cross-sections, because different individuals are observed in different time periods. Technically, if the same individuals $i$ are observed over time, then you can estimate this model with either state or individual fixed effects. In fact, the estimate of $\delta$ should remain unchanged. Again, this assumes the individuals $i$ are nested within $s$ and we observe each $i$ before and after the policy. If, however, the $i$'s move around a lot, or you observe completely different $i$'s within each time period, then you should go with a model that looks at the group differences over time.
The authors use the term "nesting" to make clear that you need panel data to estimate the foregoing model. However, evaluators also use the term "nested structure" in settings where they have repeated cross-sectional data. All they're really saying is there is some multi-level structure (e.g., kids in schools, patients in hospitals, etc.), where the $i$'s are contained within the higher level entities. Thus, it's not uncommon to see this term used more generally to describe a particular type of hierarchical structure.

"... is fit so that data comprising a panel of individuals is observed across the time for which individuals are nested within treatment group."

The author is communicating to you that you could estimate the effects of a state level policy using individual fixed effects so long as the same individuals are observed before and after the policy. Obviously, this means you need individual level panel data. The individuals within each state all become exposed to this policy, and we must observe them before and after some treatment of interest.
In a setting with repeated cross-sections, we should estimate the following,
$$
y_{ist} = \gamma_s + \lambda_t + \delta D_{st} + \textbf{z}_{ist}\beta + \epsilon_{ist},
$$
where $\gamma_s$ denotes state fixed effects. Here, the individuals $i$ are still, technically, nested within $s$, but only within that time period. We're repeatedly drawing new individuals $i$ in each successive time period. But remember, a difference-in-differences estimator is fundamentally a within estimator. If we sample different $i$ in each successive time period, then we can't model the "within-person" changes over time. We can, however, model the "within-state" effects using the individual level outcomes, as the policy is well defined at this higher level.
