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I don't want to play with physical laws in a frivolous way.

Assuming that the nature of matter and energy is the same, can a high density of highly energetic photons produce a gravity force?

We do know that radiation is affected by space-time distortions, or in another way "feels gravity". Why do photons can (or cannot) produce a gravity field?

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First of all, photons are not a pure form of energy. They are particles without rest mass, which means that they travel at the speed of light. Energy is a property of physical systems, the statement that something is energy makes no sense.

To answer your question: yes, electromagnetic radiation/photons contribute/s to the curvature of spacetime and therefore to gravity. The Einstein equations of general relativity related the curvature of spacetime to the stress-energy tensor, which contains not only mass but also energy. This also includes the energy of a photon.

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