[drc]

It's safe to guess it's real. I've never heard of reproductions of the early baseball mirrors, and if there were reproductions you'd see them for sale often and the existence of reproductions would be well known. There's a big upfront production cost to making things in metal or plastic, and the only way to profit is to make lots of them (lower the cost per item). That's why you see those repro Ted Williams Root Beer signs all over the place. Also, a pocket mirror is a neat antique artifact, but isn't the type of thing that would be reproduced for modern mass consumption. Pins, buttons, pennants and tin signs sure, but most people on the market today would say "A baseball pocket-WHAT?!?!" Like trying to market baseball spats to the masses.

I would guess the team image side is celluloid, which was an old time plastic no longer used and often used for mirrors and buttons. It should feel and sound like plastic, maybe a bit like enamel. Celluloid was often ivory color as here, as it was commonly made to mimic ivory.