Pre-modern people, internationally tended to count the passing years with a count of a specific season. In my Introduction to Folklore - with this subject borrowing heavily from the Introduction by my mentor, Sven S. Liljeblad (1899-2000) - the following excerpt addresses this:

For many pre-industrial cultures, seasons were far more important than the concept of the 365-day year. To count the passage of years, people commonly referred to the number of a specific season that had passed. They might, for example, count years as winters. Liljeblad writes that “when asking somebody about his age, a Bannock Indian says, ‘How many winters ago did you become a human being?’” In some Slavic languages, the words for “summer” and “year” are the same when considering numerous years, which were expressed by the number of summers that had passed. The Russian word "lyet," meaning both the season and the year, is a good example of this.

The lunar cycle was also extremely important to pre-industrial people, who often paid little attention to the solar calendar. Thus, they could count the passage of time with the phases of the moon and the passages of seasons. Priests or government administrators usually led the way toward adopting a solar calendar. This was true during each of the Classical periods of civilization in Egypt, Mesopotamia, Greece, and Rome. The Church also followed this model, exporting it with conversion. Nevertheless, changing to a solar calendar was a slow process, and peasants only reluctantly adapted to this new way of viewing the passage of time.

Your example of Tolkien shows the old master's understanding of pre-modern society. Tolkien no doubt understood that a fixation on dates and specifically on years dating to a specific time in the past would have had a modern "feel" to it. He allowed some of this sort of thing to be present in his novels, but it was kept in the background, which would have been in keeping with a medieval peasants point of view: for the most part they would have been aware that someone was keeping track of the number of years since the time of Jesus, but they might have had only a vague idea of how many years it had been. A personal count of the passage of seasons would have been a more immediate way to keep track of time.

Recently, I moved to a rural farming community, and I have noticed that there is a tendency to measure the lives of old farmers in the number of harvests they had during their lives. Of course, everyone here is keenly aware of the year and the calendar, but there is still that fall back; I hear people say things like "that guy there has seen 67 harvests." This refers to his career - that would mean that he was older than 67 years, but they still use the cycle of the seasons that is so important to them to measure the passage of time.