I want to add on to @StatsStudent's answer.
There are a few ways to control for a confounder. One is by including that confounder in a well-formed propensity score and conditioning on the propensity score. Another is conditioning directly on the confounder. If $Z$ is a confounder, then generally it belongs in the propensity score model. However, it might be possible to get a well-formed propensity score model without $Z$, in which case it would be sufficient to control for $Z$ by stratifying on it (i.e., including it in the outcome regression model, possibly interacting it with treatment). Indeed, Eeren & de Rooij (2015) found through simulations that even if treatment assignment depended on an effect modifier $Z$, it was preferable to exclude $Z$ from the propensity score model but include it in the outcome model as a treatment effect modifier, just as the authors of the paper you're referencing did.
In general, this is a risky move, and it's theoretically confusing why Eeren & de Rooij found what they did. It may simply be an artifact of their simulation. That said, if one has confidence that $Z$ is not a confounder but is an effect modifier, then there is no requirement to include it in the propensity score model. It goes in the outcome model to estimate the effect modification relationship, but it isn't necessary to additionally control for it through the propensity score.
A major limitation of using the propensity score in a logistic regression model instead of performing matching or weighting is that the estimand is not the average treatment effect in the population. Odds ratios are not collapsible, which means the conditional odds ratio (i.e., when other variables are in a regression model) differs from a marginal odds ratio (when no other variables are in the regression model). Including the propensity score in a logistic regression model for the outcome means the interpretation of the treatment effect is the odds ratio conditional on the propensity score (i.e., for a population held at a given propensity score, what would be the ratio of the odds of the outcome event were the population to be assigned treatment vs. assigned control). Instead using matching or weighting to adjust for confounding and only including the treatment in the outcome model would yield a marginal odds ratio (i.e., for the entire population, what would be the ratio of the odds of the outcome event were the population to be assigned treatment vs. assigned control). The latter is typically what researchers want, which is what motivates using these methods over including the propensity score in the outcome model.
So, the authors took two risks here: first, by not including the effect modifier $Z$ in the propensity score model, they assumed $Z$ was not a confounder or that the results of Eeren & de Rooij justified its exclusion even if it was a confounder. Second, by using covariate adjustment for the propensity score instead of matching or weighting on the propensity score, they assumed the conditional odds ratio (what they got from the regression) was the same as the marginal odds ratio (what they probably wanted to estimate). Many other assumptions were made as well that could severely invalidate the results: they assumed no effect modification by the propensity score, they assumed the correct functional form for the relationship between the propensity score and the outcome, they assumed they had a well-formed propensity score, the list goes on... (and some of these were mentioned by @StatsStudent). In general I would be very wary of the results of this analysis.
Eeren, H. V., & de Rooij, M. (2015). Estimating Subgroup Effects Using the Propensity Score Method. Medical Care, 53(4), 8.