Neither, it's both.
You are correct in that strenuous workouts cause damage to the individual muscle fibers and that they need time to repair. When they do repair they are thicker and stronger than before, this is what bodybuilders do to get their desired results. It is called hypertrophy. However your coach is right in that it is not always necessary to rest a day because not every workout damages the muscle fibers to the point that they need rebuilding. This is why someone can do hundreds of pushups each day without really needing to take a day off, I know because we never got a day off in basic training. So it really depends on the type of workout you are doing.
One thing you might try is to vary the intensity of each workout so that you have Heavy, Medium and Light days. I'll take your benchpress number as an example. I'll be using a total of five sets per day.
Mon/Hev - 90, 95, 100, 105, 110
Tue/Lit - 90, 90, 90, 90, 90
Wed/Med - 90, 90, 95, 95, 100
Thu/Lit - 90, 90, 90, 90, 90
Fri/Hev - 95, 100, 105, 110, 115
The next week add 5 pounds to each set, keep adding 5 pounds per week until you cannot finish a full five sets. When this happens go back to the weight you were using three weeks before and build back up to your sticking point. This should give you enough recovery time between heavy days and give you ample time to practice your form on the light days. You could also do your heavy day on Wednesday and move the medium days to Monday and Friday.