The following excerpt from my Introduction to Folklore may help:

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 particular seasons that had passed. They might, for example, count years as winters. Sven Liljeblad (1899-2000) 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.

In Tok Pisin (a pidgin spoken in Papua New Guinea), the customary way to ask a person how old they are is hamas krismas yu gat?, which is derived from the English "how much Christmas you got?" (how many Christmases/winters have you seen?).

This is a popular item included in introductory linguistics textbooks when discussing language contact (pidgins and creoles), because it's fairly transparent. It's in the textbook produced by Ohio State that we use when teaching intro linguistics courses.

This is a question that might best be answered by flipping it. In reality the "x moons old" or "x winters/summers" old is absolutely standard, the real question is how in the world did it become the case that we don't still do everything this way. The first part of the answer is that as languages go on they can often develop new words, and as such old concepts get divided out among new words. For example, moon and month come from the same word, and "year" used to be a word for "season". Once we had two words for the same thing, gradually the meanings diverged.

The other part of the answer is that as we become more and more urban and have stronger and stronger social systems to protect us from the vagaries of the seasons, it becomes possible to think of time in ways that abstract it away from nature. Within Europe, by far the largest driver of this sort of change would be the church and its accompanying calendar of feast days and holy days. this introduces an entire framework for talking about time that is not linked to the harvest/fishing/planting/summer/winter cycle. This pattern is in turn similar to what governments have often done - for example the Julian Calendar was an attempt for governments to put some sort of control on time, and to link people's understanding of the passage of time to civic responsibilities and powers.

When we look at cultures without such a history, time can be measured very different. For example within Cree and Michif, we can ask someone tanimaikohk ee-atipiponiyen? - how many winters have you passed? - but we can also ask "how many summers" and "how many years?" Since our lives were strongly connected to seasonal labour, seasons were very important. Within the Nuxalk language there are some twenty or thirty seasons, many of them overlapping, with names like "the time of fernroot digging" "spring-salmon season" "berry picking season" and so on. These all corresponded to the relevant time of the year, with some of the words being abstracted one level to refer to the specific moons. In the middle of the winter, when all anyone did was tell stories and dance and have ceremony, was the ceremony season, which lasted 50 days. Within this system, abstraction only went one step, despite the population having lived in situ since the Pleistocene. The coolest part for me here is that distances were measured in days travel, month's travel, and years of travel!

What I'm getting at here is that "X years old" is an abstraction, as is all language, and that certain types of abstraction, especially abstraction that allows us to begin to measure time entirely in largely arbitrary units such as government established units of months, is not something that happens easily or quickly. People tend to prefer to use 'natural' units unless their world has been modified to the extent that 'natural' no longer plays as meaningful a role as the 'unnatural' units, through the pervasiveness of religious, governmental, or similar systems.