A particle's interaction (with anything it can interact with) can be thought of as it making a measurement of the physical quantity associated with the interaction, (e.g. electric field in case of the interaction between charged particles) and acting accordingly.
When we use the term "particle" with "interaction" we are talking of elementary particles and we are in the framework of quantum mechanics. To make a measurement, yes,it is correct that an interaction between elementary particles has to take place. This is the table of elementary particles, and at the bottom line it is interactions between them that form the world we observe in the microworld, and collectively the world we observe in the macroworld we live and move in. The macro world emerges from innumerable interactions of the micro world constituents.
A field is not an elementary particle. A field in second quantization manifests elementary particles according the operators operating on the field, so it is not as simple. To measure the electric field a huge number of photons are involved and it is not a simple example as you think.
To make a measurement one first needs a frame to make measurements "in" (a Lorentz frame in light of relativity).
A lorenz frame can be any frame correctly defined. All interactions observed in that frame can be transformed to other moving frames, but we tend to work with lorenz invariant quantities so as not to worry about transformations,
Assigning such frame to a photon appears to me, to be problematic in the sense that constant velocity of a photon in any inertial frame implies a photon in its own frame having a velocity c (the speed of light).
A photon has no rest frame, is what you mean. There is no reason for it to have a rest frame other than the prejudices we carry from the macro world. The lorenz transformation assures that the mathematics of any frame are correct for photons too.
Now if we assume that no such frame exists for a photon, photons cannot interact with an other one.
Two photons define a rest frame , because two photons have an invariant mass which has a frame where all momenta are zero. Example the pi0 decay to two photons.
Can we explain in this way that photons do not interact with each other, or more generally particles moving at velocity c do not interact with similar particles?
No, we cannot, because they do interact through exchanges of virtual particles at higher orders in the mathematical expansions of the solutions of the specific problem, and thus with small probabilities. Gamma gamma scattering experiments exist, and gammas are high energy photons. They are even talking of gamma gamma colliders,