Just to add my two cents: ¿does someone realize that asking changing the "reference" calendar according to civilization is akin to asking changing the UI language depending on the civilization you play with?. Of course, it would be inmersive, but ¿would you manage the menus written in arabic when playing as Saladin or in greek as Pericles?.
Said this, the more options the better, as games are localized, the calendar could be as well. Currenlty this is nearly no space problem, therefore, I'm all in favour of having an UI option allowing you to change to the calendar you prefer, even asfking for it to be civilization-based. But, as an UI option, it makes sense to be linked to the player, not the game.
That is the reason as well to have the reference calendar. For the player to compare his "achievements" against the history he has been taught (of course, this is still not fine-tuned in Civ VI: but ¿how would be know if not by the calendar?). In that sense, it would seem more reasonable that the calendar changes according to the localized version (i.e. use the Chinese calendar for china release) - and to avoid issues, as soon as it is possible, make it available the option to "switch" versions (as it is now with languages).
Finally, I'm no native english speaker, neither living in any english-speaking country, so the discussion BC/AD or BCE/CE seems quite sterile to me. As some pointed out is just "dressing" a christian based calendar for it not to be seen as christian based calendar (but a common convention), yet still being a christian based calendar... (in other words, it looks like one of these cases were political correctness shows it can do quite pointless weird things).