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.
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, Sunday December 4; Quartidi 14 Frimaire, An CCXXXI; 13 Azar of 1401 Solar Hijri; Reiwa 4; 7531 Anno Mundi.
Welp. I wrote this originally intending to just link to an existing answer or two with a bit of commentary, but it seems to have become a mini answer in its own right... Anyway! Most commonly (especially in the sorts of societies that had use for scribes and written calendars and such), using regnal years.
There are a number of previous answers on Askhistorians about regnal chronologies (for example, this Monday Methods by u/Bentresh discusses the chronologies of the Bronze Age Hittites and Egyptians). That post actually includes a direct quote of regnal year dating in action, with an Egyptian inscription dated to the first day of the third month of Akhet (i.e., of flood season) in Year 11 of Amenhotep. As Bentresh noted, stitching regnal years together into an actual context that lets you understand what they mean requires either the familiarity of having lived in or close to those times or else the ability to consult a list of rulers and reign lengths (if I, an American, told someone I was born during the Reagan years, was a kid during the Clinton years, and a teenager during the Bush Junior years, I don't think most people would find it hard to understand me, but if I told them my grandmother was born during the second government of Ramsay MacDonald, they might have to go looking for a reference).
Among other functions, the king lists produced by ancient scribes could be consulted to contextualize regnal year dates if the reigns of the rulers in question had passed out of living memory.
Similar systems prevailed in many other ancient societies - a resident of Achaemenid Susa might write the date as the sixth year of Cambyses and a Neo-Babylonian might record the year as the twelfth year of Nabonidus. Variants of the same system prevailed in ancient Mediterranean republics and democracies, which assigned the year-naming function of kingship to other officials; if you lived in Athens, you could tell someone what year something happened by giving the name of the year's archon; in Rome, you could date by consulships (it happened when Caesar and Bibulus were consuls).
In fact the most widespread modern year system, CE/BCE (AD/BC) is itself also a variant of regnal years, although I don't know that most people literally conceptualize the current year as the two-thousand twenty-second one that belongs to anyone's reign. That's what anno domini means, though (common era/before common era being the generic rather than the religion-name-brand version).
And as an ending anecdote, the guy usually blamed for coming up with the idea of dating from the birth of Jesus had to record the date in a fashion that was more familiar to his contemporaries so that when he wrote that Jesus was born 525 years before the present, future readers would understand when his present actually was. So he left them a note explaining it in the terms they would understand: p.s., guys, I'm writing this in the year Flavius Probus Iunior is consul.