The term "group" is a bit of a misnomer. The authors specify a model on page 457 to show how to estimate the "generalized" difference-in-differences equation in practice. Whether to use fixed effects at the country, state, county, zip code, school, or individual level is completely context-dependent.
For example, if a law is passed at the state level and we're interested in learning about its effects on state level outcomes over time, then I may specify $a_g$ as $a_s$, which denotes state fixed effects. Likewise, suppose a school lunch program was administered at the school level in one major U.S. county. Some schools within the county received the school lunch program and others did not. In this setting, $a_s$ denotes school fixed effects. And finally, suppose a police crackdown was instituted in one metropolitan city but it was administered at the district level. Some districts cracked-down harder on gun violence, but due to budgetary concerns the intervention could not be extended to all districts citywide. Some of the "other" unaffected jurisdictions may serve as controls. In this case, $a_d$ would denote district fixed effects.
The notation isn't as important as you actually describing to your audience what the parameters actually mean. The "group" fixed effects is technically referring to the panel unit. If the panel unit is firms observed over time and the policy impacts some firms and not others, then it is customary to estimate firm fixed effects. Suppose we observe 200 firms over many years. Only half receive some intervention midway through the panel. Technically, we only have two groups (i.e., treated/untreated), yet we wouldn't instantiate a simple treatment/control dummy. Instead, we estimate a full series of $N - 1$ effects. This results in 199 firm effects.
This is often confusing to read for the first time because we actually don't care much about the "group" in this more general setting. All we have is units $i$ (i.e., customers, universities, firms, precincts, counties, states, countries, etc.) equaling 1 if treated at time $t$, 0 otherwise.
I want to conclude with a quote from page 457 of the paper you referenced above:
In practice, researchers estimate the treatment effect parameter, $\delta$, using fixed effects regression models; they simply regress the observed outcome on the treatment variable and a full set of group- and time-fixed effects
The term "group" is used generically in the same way "time" is used. What are time fixed effects? What does time represent? Again, it's context-dependent! If we observe state level outcomes over 10 years and a state level policy, then we often say we're estimating state and year fixed effects. Similarly, if we acquired weekly gun fatalities in districts pre- and post-intervention, then we'd use district and week fixed effects. The purpose of replacing "group" with "district" and "time" with "year" is for clarity.
