The main problem is that photons only can interact with charged particles, but photons have no charge, so the photons don't interact (directly) with photons.
The gluons that are the bosons of the strong force can interact between them, so it's not a problem of bosons vs fermions. In particular, gluons have color that is the equivalent of charge for the strong force.
The idea is that the photons can sometimes create a virtual electron-positron pair that is very short lived [1], and the other photon can colide with the electron or positron before they annihilate. This is very rare, and you can ignore it unless you have a really huge amount of photons flying around.
[1] Insert here a technical remark about "virtual" and "very short lived". It's more complicated. Take this as a metaphor to hide a lot of math.
The main problem is that photons only can interact with charged particles, but photons have no charge, so the photons don't interact (directly) with photons.
The gluons that are the bosons of the strong force can interact between them, so it's not a problem of bosons vs fermions. In particular, gluons have color that is the equivalent of charge for the strong force.
The idea is that the photons can sometimes create a virtual electron-positron pair that is very short lived [1], and the other photon can colide with the electron or positron before they annihilate. This is very rare, and you can ignore it unless you have a really huge amount of photons flying around.
[1] Insert here a technical remark about "virtual" and "very short lived". It's more complicated. Take this as a metaphor to hide a lot of math.