EBII seems to use the "alternate" piety system from the Britannia Kingdoms campaign, as described
here. In the Britannia campaign, M2TW's "piety" became "management" and "religion" became "culture".
In EBII, "management" became "influence", but it seems to work the same. That is, a city governor's influence increases the faction's culture in a city (the greater the influence, the faster the change)
without causing additional unrest from other factions present in the city; and it increases income, first by decreasing corruption, then by adding administrative income.
Presumably, culture is also influenced by the cultures of neighboring provinces, as described
here.