They couldn't have matched the time of infection, since not everyone in the study was (re-)infected during the study period. The point of the study was to see if there was a difference in the infection rates, so just picking from the people who got infected would have completely distorted the results.

But they would have arranged the study such that both people in the matched pair were considered for exactly the same time period. E.g. if the vaccinated person in pair was fully vaccinated on April 20th, they would have started counting re-infections for the previously infected person of the pair from April 20th too.

Local infection rates should be the same, since the pairs would be matched from the same geographic area and the study contstruction ensured they were considered for exactly the same time period.