Do you mean there are two kinds? Or if you swap any two?
This is why two fermions can’t be in the same state: if you swapped them, you’d be back where you started, but that somehow also had to phase shift 180 degrees, and the only complex amplitude that is the same as its own negation is zero.
In contrast, anything phosphorescent most definitely absorbs and re-emits, sometimes hours later. Imagine glow-in-the-dark pigment.
Physics is all about choosing an appropriate model of the world that captures the detail you need without being more complicated than needed. Is a person on a playground swing a pendulum, or is it a bag of molecules sitting on a flexible piece of rubber suspended by a bunch of rusty metal links from a flexible steel bar that is, in turn, supported by many geological layers in the Earth’s crust?
This part is philosophy, not physics. Photons are “indistinguishable particles”. You cannot tell two photons apart. This has a rigorous meaning: if you swap two photons, the state of the universe is unchanged. (This is in contrast to fermions — if you swap two fermions, the state of the universe has its phase shifted 180 degrees.) If photons we’re distinguishable, then swapping two photons would swap them, and the resulting state would be different. The matters from a statistical perspective. Look up Bose-Einstein statistics if you’re interested.
As for whether photons are absorbed and re-emitted or merely reflected, this surely depends on the surface on question.