I liked the video and the kinds of questions it raises.
In raising the questions, there is (as the video notes) always the historical perspective and the game perspective. Periodization is a huge issue within the field of history. A game just needs phases it can go through. In Monopoly there's the buying properties stage, the house stage and the hotel stage. As a mere strategy game, a game like Civ could just have Age 1, Age 2, Age 3, or just a time-line with no ages designated (as the streamers point out, in earlier incarnations, the divisions didn't have much bearing on the game in their own right, separately from their being the phases at which particular technologies came in). Since Civ is historically-themed, there will be some correspondence between their names for their ages and terms that historians use, but, again, those names are an object of controversy within the field of history, so there's not likely to be consensus anyway on the historical side of things.
On the game end, as the video remarked, they just want the three act structure of storytelling. Antiquity means Old, Modern means now; they need some name for the second, but "Middle" is problematic, since it has a place in a different tripartite naming structure, and strong associations as to time period as a result, that don't entirely match this game's middle. So they picked something that is 1) characteristic and distinctive about that broad era of human history and 2) one of the Xs in 4x games.
It's a good historical question what would make for a better name for Age 2 on these terms. If we know that the period is roughly 800-1800 (I date from when one recovers from the crisis and starts the next era, not from when Antiquity ends), you would have to think of something that is true 1) globally of that period, and 2) as distinct from the other two. Reaching out toward a global level of connectedness strikes me as filling the bill about as well as anything I can think of, and, given the time-frame they wanted to carve out for their Age 2, the idea of "exploration" is capacious enough to take in earlier Norse and Polynesian efforts as well as the colonialist explorations of Western European countries.
Even if every civ got its own calendar, there would have to be a master-calendar that coordinated all of them, again if only for game-play reasons, Since the game of Civ is fundamentally a race, you need something that can mark various civs' progress, relative to one another, in that race. Chutes (or Snakes) and Ladders is a race; all of the players need to be competing on the same grid, and with the same limits on how much they can advance in a turn, for there to be any kind of measure of who won.