Are we not living in the modern era anymore?

[Belacqua]

Isn’t the word “modern” synonymous with the word “contemporary”? If that’s the case, then we are always in the modern era, and we would need to come up with a historical name for the era that ended at the turn of the millennium.

This can get confusing, depending on the field you're talking about. 

As you know, "contemporary" means "at the same time as," so we can say that Leonardo was contemporary with Michelangelo. (I understand that's not the sense you're using it here.) 

In the arts, "modern" generally refers to around the first half of the 20th century, with no clear end point. Though you might point to Andy Warhol or somebody who was at the end of the modern era, or even as putting an end to it. "Modern Art" tends to have a fairly clear set of goals and theories behind it, which are no longer so popular. If you go to a Museum of Modern Art it will have works by a lot of dead people -- Picasso, Kandinsky, etc. A Museum of Contemporary Art will have mostly works by living people. Although as time goes on a lot of those are dead, too, which makes it confusing. Keith Haring, for example, is probably Contemporary, though he died decades ago.

As time goes by, anachronistic words stick. The Pont Neuf -- "New Bridge" -- in Paris is the oldest bridge. Art Nouveau -- "New Art" -- is now old.

It's just as confusing in Japanese, where 現代 (present age) is different from 近代 (near age). Picasso is present age while Banksy is near age. If I were the Emperor of Language, I would start this over from scratch.