For these reasons, it seems reasonable to ask whether these findings can be replicated using another dataset.
This paper didn't pass peer review nor was it published by a journal. The likely reason that this paper couldn't be published was because it was conclusion-focused - it was written to achieve a predetermined conclusion, to report that there was no different in charitable giving.
Quoting from the paper we see their first steps do show a difference even with significant controls employed:
While the magnitude of the effect decreases once we control for income and then church attendance in the second and third columns, Republicans still give 85% and 110% more than Democrats, respectively. In the fourth model where we control for both income and church attendance, Republicans donate approximately 34% more than Democrats. Although the giving gap between Republicans and Democrats shrinks substantially, we continue to find a significant difference. While the substantive difference between Democrats and Republicans in the final model is similar to the previous model that controls only for income and church attendance, the result is not statistically significant. Thus, using partisanship instead of ideology provides suggestive evidence that right-wing attitudes are associated with charitable giving.
A peer reviewer is going to be put off by all of the hoop jumping that is going on in this paper in order to find ways to drive down the difference to zero between Republicans and Democrats. They've controlled for the actual significant variables of income and church attendance and still found Republicans more generous.
It's sickening to read such politicized research. Thankfully the peer review system works and this paper never made it to a journal.