I know it sounds silly, but the thought popped into my head. Have we as a society been following the yearly calendar since the year 0, for example, did those in the year 1610 say the year was 1610 or did this in the year 600 say year 600? Or was the current year counter adopted later on?
No, we haven't. There's lots on this in our FAQ:
https://www.reddit.com/r/AskHistorians/wiki/language#wiki_calendars