#with some micro-eras specific to us like the Progressive Era (1890s through 1920s I think?)

[marzipanandminutiae]

Do you think they'd call this period in English history the Charles era or would it be more like the Post Second Elizbethian era?

Unclear. It didn't always go by monarch's name, in the past- sure, you had the Victorian era and the Georgian era before it, but you also have what we now call the Restoration era, and the sub-era of the Regency within Georgian, and the Tudor era named for a royal family rather than a specific monarch- though I have heard the term "Henrician" batted around for Henry VIII's reign. I still have to keep looking up what "Jacobean" comes from, because I always forget the exact explanation (James in Latin is Jacobeus, and the king was James I). I've also heard terms like "Thatcher-era," and of course after Edwardian you get periods named for historical events or attitudes on both sides of the pond: WWI, the Roaring Twenties, the Depression, WWII, etc.

(Shoutout to Regency for being extra-confusing: the literal English Regency was in the middle of stylistic/social period we use the term for. It was 1811-1820, but the aesthetics, media, and manners we think of as "Regency" span roughly the 1790s through 1830.)

Charles I's reign was called the Caroline or Carolean era, also from the Latin version of his name. So maybe, going by monarch's names, this is the second Caroline era?