We do not use the Mayan calendar for the current year in any way, shape, or form... I'm not sure where you got that idea. Perhaps you mean Babylonian?

The use of the annual passage of the Sun as a primary means of determining the large-scale passage of time is something common to pretty much every agricultural civilization I know of, because it corresponds with the seasons, and if you don't learn the regularities of the seasons then you're going to be a pretty dead agricultural civilization. That does not mean you have a perfect calendar (so the seasons start at more or less the same time each year and don't "drift") or don't have variations (e.g. lunar calendars) or don't sub-divide the year in all sorts of different ways... but knowledge that Sun has an annual transit is far more ancient than the Roman Empire.

As for Ancient Rome, they did use years as a measure of time. Their specific dates and calendars varied by the time period. But they had a system that used years, months (ten per year), and days. This eventually became superseded by the Julian calendar in 45 BCE, which was the standard in much of the West until the Gregorian calendar reform in the 16th century. So for all intents and purposes in 189 CE they had a very similar calendar to the one we have today (though their final year count was different; the BC/AD realignment happened several hundred years later).