Apart from being a Russian custom, it's also a Jewish custom to write down father's name. It's used for a clearer identification of an individual. However, in some records the patronymics were not written down, or they were written down for men, but not women. I've seen birth records where only the father's last/first name were present and in the very next record for another child they wrote down father's last, first and, patronymic and the same for the mother of the child, including her maiden name. This is due to lack of enforcement of rules and a less informal attitude towards the information.
To answer your question about patronymics, it's not about someone using or not using a name. Everyone had a patronymic, it's just an answer to "what's your father's name, we need to write it down together with your first and last names and your age". Patronymics were required because Jewish people in that part of the world generally didn't have last names prior to the partition, yet they needed to be identified somehow not just by the tax collector, but also by others in the community. This was also true for other non-Jewish subjects - commoners didn't have last names up until it was required by law. If there are a few dozen Shevels, how do you tell them apart? You end up with Shevel son of Yankel or Shevel son of Hava, but also nicknames Shevel the readhead, Shevel the baker, Shevel from Charkasy etc. Eventually, some of these turned into last names.
You said that you googled Polish names, but even though this area was once a part of Poland, majority of the population in the area was not Polish, it was ethnic Ukrainian. In both Russian and Ukrainian cultures (and all other Slavic countries) it is also common to address someone with their first and patronymic as a sign of respect. A peasant would address his landowner boss this way, but the courtesy was generally not returned the other way around.
In 1897 in Zvenigorka (just the city) Ukrainians were 49% of the population, Jews - 38%, Poles - 2%. In the Zvenigorodka uezd/county (not counting the main city) - Ukrainians - 92%, Jews - 8%, Poles - 1%. The rules and laws of Russian empire had been in effect for over a hundred years by 1906 and the Polish rules of the 18th century were no longer relevant, just like laws of Mexico were not relevant to Californians in the 1950's, 100 years after the territory became part of the US.