However, at the quantum level, identity is not bound to space or time. When you split a photon into an entangled pair, those "two" photons are still identical. It's a bit like slicing a flatworm into two parts, which then yields (we think) two separate new flatworms... but they're actually still the same flatworm.
Experiments like this are surprising precisely because they break our assumption that identity is bound to a discrete object, which is located at a single space, at a single time.
I remember seeing that on SNL in the 80s; it has always stuck with me.