No, I still don't think that's right. I don't think the volume has anything to do with it.
A skinny road tyre is designed to hold much higher pressure (up to 140psi), because a skinny hard tyre is mechanically more efficient; smaller contact patch, much less rolling resistance. If you factor in the additional pressure caused by the riders weight, and sudden spikes caused by bumps etc, then it would be safe to assume the tyre and rim are designed to withstand even greater pressure, maybe 200psi+.
A fatter, larger volume tyre is designed for comfort and shock absorption, rather than optimum mechanical efficiency, so lower pressures are sufficient. This would surely allow for the greater bump forces suddenly increasing the pressure within. If you land from a jump, then the pressure inside the tyre could spike from 40 to 80 psi, or perhaps even more. So, the tyre and rim would surely be designed to withstand higher pressures than the indicated guide. So, I suppose in theory you could have a rim of the same dimensions/materials/proportions as a road tyre, and a fat tyre which could withstand the same pressures as a road tyre. It just wouldn't be very comfortable off road.
I'm at a loss as to how increasing volume somehow would put more strain on tyre and rim, if the pressure remains the same.