I noted that today, 21 April, is the traditionally accepted anniversary of the founding of Rome, according to the Romulus and Remus legend. The year given was 753 BCE...
Understanding that the foundation myth is just that, what would an ancient Roman say the year to be if I asked him for the date just a few years after 21 April, 753 BCE? And what about just before and after Ceasar's switch to the Julian calendar?
Same question goes for Greece. If we know that the Battle of Thermopylae took place in 480 BCE, what date would Leonidas have given for the current year if we had the opportunity to ask him just before he made his final stand?
The Christian BC/AD calendar is just one amongst many possible systems of year counting. Various other alternative calendars exist, as we can see from the appropriate section of the FAQ, Other Dating Systems under Calendars, if your browser doesn't automatically take you there. Most relevant to your question, I shall add to that a post by u/Iphikrates, which looks at how the Greeks and the Romans expressed their dates.
Indeed, BC/AD is simply a regnal calendar that refuses to change eras like normal regnal calendars. Multiple other calendar systems remain in use today. Japan still uses a regnal calendar for official purposes.
Posted on this day, Thursday 22 April; Duodi 2 Floréal, An CCXXIX; 31 Farvardin of 1400 Solar Hijri; Reiwa 3; 7529 Anno Mundi; the fourth year of the 699th Olympiad; the fifth year of Rodrigo Duterte being President.