There is more to being antimatter than just having the opposite charge. The spin of the particle also matters. For a particle to be its own antiparticle, it would have to have spin 1/2. All elementary fermions have that property, but not much else.
Of the 2 classes, fermions and bosons, only fermions can be their own antiparticles. Bosons are defined with having an integer spin, so they can never have spin 1/2. Of the fermions, none are known with neutral charge except for neutrinos, and we're not sure if those are Majorana particles or not.
Photons, as you mention, are bosons, with spin 1, so they can't be their own antiparticle.
Of the 2 classes, fermions and bosons, only fermions can be their own antiparticles. Bosons are defined with having an integer spin, so they can never have spin 1/2. Of the fermions, none are known with neutral charge except for neutrinos, and we're not sure if those are Majorana particles or not.
Photons, as you mention, are bosons, with spin 1, so they can't be their own antiparticle.