Cantilever Power Hanger I have bought an old mtb with cantilever brakes. Front has regular yoke and works just fine. 
Problem is in the rear, where there's a Power Hanger bolted into the frame just above the tire. It looks like that: 

only not on the fork.

Brakes are very soft, movement of pads is uneven, they are unable to lock the wheel no matter what.

I'm wondering if this hanger is properly mounted, should it move while braking or should it be bolted tight?

I can't imagine how this device would work. I already found that it's based on a bell crank but I can't see how it applies to braking. How, physically, would movement of one arm cause the other to get closer to the rim?
It needs to be bolted tight to remain attached, but the hanger is mounted on a collar so is always free to rotate as it’s designed to. 

So yes to both, when it is bolted tight it is still free to rotate. 

Normally a brake housing fits into rigid frame stops. These stops resist the force of the outer cable so all the cable force pulls on the brake where the inner cable is clamped. In this case, because the power hanger can rotate, when the cable contracts, the right brake arm moves as normal but the power hanger moves too (clockwise). This pulls on the secondary cable attached to the left brake arm, so both pads move towards the rim.