As an aside, worth pointing out that the "BCE" terminology meaning "Before the Common Era" is a very recent change from "BC". When I was at school in the '90s it was very definitely BC meaning "Before Christ".

Now that international commerce has kind of forced the whole world - much of which isn't Christian - to use the Gregorian calendar and a common reckoning of the year, BC has changed to BCE to be more inclusive and agnostic to religion.

I'm not exactly sure but I would suggest this shift has happened in the last 10 years.

Adapted an older post of mine in r/history:

The Ancient Greeks used a variety of systems usually based on public figures who were in office at the time. Athens, for example, was officially governed by a group of nine archons, who were selected by lot from the richest citizens and held various public offices for the duration of one year. One of these archons was the Eponymous Archon (the title literally means "name-giving leader"), whose name was used to indicate the year in which he served. Greek historical texts will often refer to who was archon at Athens in order to indicate when something happened; inscriptions often mention something along the lines of "...when X was archon" to show when they were set up.

However, this measurement of years would only really mean something to those who knew who was archon at Athens, and other city-states had different ways of counting years. To demonstrate just how varied these systems were before the general acceptance of the BC/AD (or BCE/CE) count, here's how Thucydides dated the start of the Peloponnesian War:

2.2 The thirty years' truce which was entered into after the conquest of Euboea lasted fourteen years. In the fifteenth year of the truce, in the forty-eighth year of the priestess-ship of Chrysis at Argos, in the Ephorate of Aenesias at Sparta, in the last month but two of the Archonship of Pythodorus at Athens, and six months after the battle of Potidaea, just at the beginning of spring, a Theban force a little over three hundred strong, under the command of their Boeotarchs, Pythangelus, son of Phyleides, and Diemporus, son of Onetorides, about the first watch of the night, made an armed entry into Plataea, a town of Boeotia in alliance with Athens.

What we use is called the 'Gregorian Calendar' and, before this, many people had their own calendars that they reckoned from.

For example, the Romans marked their years ab urbe condita, which means years since their city was supposedly founded. So 345 BC was considered 409 Ab Urbe Condita if I'm not mistaken. 409 years after the City of Rome was founded.

Likewise, even in our conventional understanding of years there was some variation. For example, the Julian Calendar was in use up until the 1500's, where the Gregorian Calendar became dominant in the European world but not entirely. For example, World War One began on July 28th as we reckon things but at the time Russia, who used the Julian calendar, would have said the war began on July 15th if I'm not mistaken.

Even today, the Gregorian calendar is not universally used. Much of the Islamic world uses the Islamic calendar, which is dated 1430-something. Here you can see some of the different calendars https://en.wikipedia.org/wiki/Gregorian_calendar on the right-hand side of the page.

If you landed in Rome in 345 BC, they would tell you it was the year of the consulship of M. Fabius Dorsuo and Ser. Sulpicius Camerinus Rufus. If you landed in Egypt, then it would be the 16th year of the reign of Nectanebo II.

"Reginal Year", or a variation of it, was very common. The current year would be known by how long the current monarch would have been in power. As /u/Iphikrates points out, the Greeks used the Eponymous Archon for that year. The Romans kept track by which two men were consul for the year. To understand their timeline, you have to have a "King List", or consul list, or the like.