In Hollywood time piece movies, sometimes actors do not say a person or child is X years old, but instead use phrases such as: he has seen 13 summers, or lived 9 winters, etc. Historically, was this a phrase that was actually used?
Your question immediately brings to mind the Winter Count of the Lakota (Sioux).
Bob Drury and Tom Clavin's excellent The Heart of Everything That Is, a biography of Red Cloud, contains some really good information about the life and culture of the Lakota people.
"Although they had no formal written tradition, since at least the early seventeenth century various bands had kept and passed down pictographic "Winter Counts," a sort of snapshot chronicle of the most important events of any given year - eclipses, raids, droughts- etched into a deerskin or buffalo hide." (Drury & Clavin, 42).
You can see a well-preserved example of one such count in the Smithsonian National Museum of the American Indian in Washington, DC. (The link here is to an image of the count).
The Smithsonian has another Winter Count in the National Museum of Natural History, which this 2015 article from Smithsonian magazine describes thusly,
Pictographic calendars, originally painted on hides, were kept by two tribes in particular, the Kiowa and the Sioux, or Lakota. They have come to be known as “Winter Counts”—so called because each year was believed to commence with the first snowfall.
So anyway all this to say, yes, there are at least some cultures that marked time by the passing of seasons like winter.
Sources
Drury, Bob and Tom Clavin (2014). The Heart of Everything That Is: the Untold Story of Red Cloud, an American Legend. Simon & Schuster, p.42
Momaday, N. Scott, (2015). "The Year the Stars Fell" Smithsonian, Jan. 2015, p.42