Do modules tend to go out in pairs or singles? Also why does the ECU only check them in pairs?

In a pair of modules, the faulty cell usually has a high internal resistance. When charging and discharging with a high current, the faulty module heats up significantly, which leads to heat transfer to the modules located nearby, which also begin to degrade.
Why is a pair of modules measured? Probably, this is how it happened historically. Engineers thought it was more convenient this way.

The test leads run from only one side of the battery, attached to one contact plate.

 

Each module is already six NiMH cells, so by monitoring module pairs, they are monitoring strings of 12 cells. With NiMH you can get away with that, because NiMH chemistry is forgiving enough.

In the lithium batteries Toyota builds for newer Prii, they do in fact have wires back to the ECU from every single cell, which means a lot more wires. In the NiMH batteries, because they can get away with fewer wires, they do, and keep cost and complexity down.