How much if this is speculation? Also, do other particles behave like this?
I didn't realise particles could be their own antiparticles, but it transpires that e.g. photons are, because all photons are neutral, not charged somehow.
However, even though a proton is its own antiparticles, two photons do not annihilate, right?
Yes, as you note, photons are their own antiparticles.
The maths doesn't have a preferred time direction, so two photons can annihilate into an electron-positron pair.
I'm not sure if this has actually been observed given how hard it is. That said, my favourite type of supernova is caused by pair creation, though I don't know the proportion of that which comes from 2-photon interactions: https://en.wikipedia.org/wiki/Pair-instability_supernova
There's also Majorna particles, but as I understand it the only known particles that are definitely Majornas are also quasiparticles:
The wikipedia article on the Breit-Wheeler process has some history of the work on experimental observations, although I don't know how accurate or up to date it is https://en.wikipedia.org/wiki/Breit–Wheeler_process
But only if their energy is high enough. So by that reckoning, photons below 511 keV don't have antiparticles, and those above it do. That's pretty weird. So maybe it's better to say that photons aren't really their own antiparticle, but they might theoretically destroy each other in some rare circumstances.
Nobody is sure, but some people think that neutrinos are they own antiparticle https://en.wikipedia.org/wiki/Neutrino#Majorana_mass I never liked that theory, but some people that know more than me about particle physics liked it.
There were some experiment using atoms that decay ejecting two neutrinos, and hopping that in some case the two neutrinos will annihilate each other. https://en.wikipedia.org/wiki/Neutrinoless_double_beta_decay . IIRC, none of the experiments found the strange annihilation, so perhaps neutrinos are not their own antiparticle :) .