Sure they do! It can be interesting or infuriatingly complicated, often both. I can give an example from Japan:

gengō (Era Name)

When a Japanese emperor ascends the throne, a name of a new era is declared. This era name is called gengō (元号) . After his death the emperor (Tenno) will be referred to by his era name, not by his given name.

For example this year is AD 2020, or Reiwa ( 令和 ) 2 (as the current emperor ascended last year).

Historical events are often referred by an era they took place.

For example 天明の大飢饉, Tenmei no daikikin is a famine that occured in Tenmei Era (daikikin means famine).天明 Tenmei, is an era lasting from April 1781 through January 1789

Historically, it was common to choose a new era names during some years of the Sexagenary cycle (a Chinese 60-year "century" for a lack of a better term), or after certain important events. One emperor's reign could be spread across multiple era's.

This practice was dropped after the ascension of Mutsuhito, the emperor Meiji. Since then, era changes only with the new emperor.

The system is complicated, but since it was uninterrupted and official practice of the court since 701 AD it is very clear and unambiguous.

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There are unofficial era names used by various institutions (such as shrines or temples), but they can be conflicting, missing periods etc. so they are less useful for referring to a period of time.

jidai (Period Name)

There are some names of the periods that are used for the convenience when talking about history

• Jōmon period14,000 – 300 BCE (named after a style of pottery)
• Yayoi period 300 BC – 300 AD (named after the neighborhood where important archeological site was)
• Kofun period 300 -538 AD (named after the shape of burial mounds used)
• Asuka period 538 -710 AD (named after a capital city)
• Nara period 710-794 AD (named after a capital city)
• Heian period 794-1118 AD (named after the capital city, now named Kyoto)
• Kamakura period 1185–1333 AD (named after the city whre Shogun's court was located)
• Muromachi period, or the Ashikaga period 1336 to 1573 AD (Named after a Kyoto neighborhodood where Shoguns resided. Ashikaga was the name of the ruling clan who were the Shoguns).
• Sengoku period, "Age of Warring States"; 1467–1615 AD. It got its name from the "Age of Warring States" in China ( 475–221 BC ). You can see there is an overlap with Muromachi period. That is because the Ashikaga shoguns were slowly losing control, and the military conflicts were common towards the end of the dynasty.
• Edo period 1603-1868 AD (named after Edo, today's Tokyo).
• Meiji era. Meiji was a name of the emperor, and from Meiji on usually gengō is used as there is less need to group several gengō together.

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Unlike the official records of gengō which are clearly defined, the jidai can be split, merged or dates moved according to an opinion of the author. For example Kofun and Asuka periods are often merged and referred to as a "Yamato period."

It is similar to how Europe has a traditional day the ancient age stopped and middle age finished, but you could argue it could be moved by centuries depending on the view, what area are you describing or just by the opinion.

If you talk to a Japanese person and mention i.e. "Kamakura period" they will know what era you are talking about, and it is a default method of referring to historical age.