I read that the interaction $hhh$ is allowed but $\gamma\gamma\gamma$ is not allowed. But both fields correspond to bosons. Why should one kind of interaction be allowed but not the other?
The QED lagrangian is even under charge conjugation, and so not even effective lagrangians of it support this term (Furry's theorem). Specifically, photons are odd under C, the transformation that reverses charges. Reversing charges means reversing the vector potentials coupling to them, maintaining the lagrangian invariant.
It is our world, not another, notional one. Just so.
By contrast, the Weak sector does not conserve C, or P, or even CP (!), so assigning a C eigenvalue to the Higgs that would constrain its couplings is meaningless.