Matching & DiD

Hi, I understand well both matching and DiD, but I have a hard time understanding the link between matching and DiD methodology used in a famous research article. Maybe some of you can help me understand it. Ill link to the article below.

First she assigns a matched control firm to each treated firm by some matching variables - I understand well how this works. NB! Control firms have never experienced the treatment in any year.

Then she estimates the regression y_it = α_i + α(c) × α(t) + α(s) × α(t) + β(Treatment_it) + ε_it.

The problem is that I do not understand how a treated firm gets compared to its specific matched control firm in this regression. From what I understand she just creates a pool of matched control firms that will be added to the data frame. Morover that the DiD just applies since she add fixed effects, and not by comparing the treated firm to its specific matched firm. Also that the treated firm gets compared to the pool of firms that just didnt experience treatment that year, both including treated firms (not experiencing treatment that year) and matched control firms.

i = firm
t = year
c = countries
s = industries
y= variable of interest (e.g., CO2 emissions, institutional ownership, etc.)
αi = firm fixed effects
αc × αt = country by year fixed effects
αs × αt = industry by year fixed effects
Treatment= dummy variable (“treatment dummy”) that equals one if firm i has issued a green bond
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Less is more. Stay pure. Stay poor.
Yeah this model is a little different then I am familiar with. I am a medical researcher not an economist, which they love these things.

I am used to the following:

y-hat = Beta_0 + Beta_1(group) + Beta_2(period) + Beta_3(group*period)

Then if observations in the two groups were matched you would just use them, not sure if a variable would be needed in a strata statement for this. May depend on the matching strategy. With another approach being, just using weights (inverse propensity scores) to balance the groups. I have not done one with repeated events or time, since that usually gets used in an interrupted time series instead to control for time based correlation instead of DiD.