The red rash or “redness” you describe is probably a variation of classical hives, and hives a very common phenomenon in the general population.. There are many “causes” of hives, some of the more common ones being: Animal dander, pollen, mold, plants, latex, plastic tubing, exercise, heat, cold, sunlight, foods (berries, eggs, milk, nuts, tomatoes, shellfish) insect bites, and mechanical stimuli (harsh massage, vibration, excessive/sustained pressure).
The possibilities suggested in your case would include heat, cold, any soaps or body washes, each of these perhaps in combination with hard, pulsating delivery of the shower water.
The preceding is far from inclusive but should give you an idea of the types of things that might combine with your shower exposures to produce the hives.
I suggest you try showering with low flow water at or near room temperature and gentle scrubbing with the most simple, non-perfumed soap you can find.