It was also convenient to have numbers, too, (especially if you don't have a good memory for history), so the Romans also used dates AUC, which is a Latin abbreviation for "Since the founding of the City", referring to the supposed date when Romulus founded Rome.
There were also various religious calendars in use; the Jews, for instance, numbered (and still do number) years from the date when the world was supposed to have been created.
The Roman calendar would have been used across a wide swath of Europe and western Asia. I understand the idea of referencing times of rulers or kings or events, but was there a bigger picture, if you will, as to how time was kept.
Given the strength of the Roman empire, its calendar was probably used or at least referred to elsewhere, just as the Christian calendar eventually became today's standards even in non-Christian countries. "The 7th year of king So and So" would have been in the year "XXXX".
Still another example would be of the Roman Republic.
And the Roman calendar most definitely influenced that later Christian calendar, which, to grossly simplify, took over the Roman year and renumbered its start. I know now that WE can say it was XXXX BC but did they have some sort of larger calender back then? I would think it would've been hard to figure out a person's age without a year system set up.
Especially when you start outliving one, two or three emperors/kings.
Japan, today, still uses dynastic numbering of years for the civil calendar -- but use the Gregorian calendar for purposes of months/days/leap years, i.e. In the historic tradition of calendars, the years are ordinals, with no year zero: 1989 was Heisei 1 but it started as Showa 64 (the Showa emperor, Hirohito, died in January).
If the question is how come the count of years from a slightly-askew estimate of Jesus Christ's birth is the standard even in many non-Christian societies, well, that's just what Christian Europe adopted as a benchmark in the early Middle Ages and it just so happened they became the global-dominant cultural bloc and it's convenient to standardize to the Big Players.