I would guess that they set up the chess board incorrectly rather than the chess board being set up differently in the 19th century.
When did "light on right" begin?

Batgirl would probably know I know right on red started in the late 60's. Oh, my bad. Don't play chess like you drive

Yeah, pretty certain it's a flub on the part of the Met...the curators probably know more about art than they do about chess...surely it was well-crafted set, and was still enjoyable to observe.

*OP taps on display case*
"This is an OUTRAGE...this chessboard is set up INCORRECTLY, and I DEMAND that it be switched out IMMEDIATELY!!!"
"Security, report to gallery seventeen!"

The checkerboard colors started in Europe around the 13th century, but judging by old paintings (e.g. "Otto with the Arrow", 1305; or a diagram in "Libre De Los Juegos", 1283), there was no specific orientation expected, or they chose "Red on Right". This was never universal though, as exhibited with Shogi which never adopted it. That said, it might be a mistake on the part of the artists.
White on Right was depicted in "The Chess Players" - Leydon, 1510; and Caxton's "Chess Book", 1474. So potentially, it become convention around that time. Of course, it might not have been mandated.
Edit: I've done more searching and found some puzzle books from 1493 that depicted Black on Right. Again, might be the artist's own vision or indicative that the color was not yet obligatory. Incidentally, "White Goes First" rule can be traced to the late 19th century, so maybe the board orientation was enforced at the same time, if it wasn't already.
For how long have chess boards been oriented the way they are, with the lower right hand square being white?
I ask because recently I was at the Metropolitan Museum of Art in Manhattan and they had a chess set from, I don't recall, maybe the early 19th century. Anyway, it was oriented incorrectly!