What property of USB battery packs allows them to simultaneously output power while receiving power?
I don't know, but this would seem to be a design choice. The simplest way to build one of these things is to have a "T" shaped connection between the battery, its charger, and the boost converter for the output. However, that means the charger can't tell whether current is flowing into the battery or to the load. This makes both current limiting and charge termination more difficult.

It may be the case that running something from the pack output causes the battery to go through lots of tiny charge/discharge cycles. This is bad for its life. I suspect that the expensive ones regard this as bad and prevent it while the cheap ones don't care.

There's also the question of what happens if you try to draw more current from the pack than the charger is supplying: you'll definitely have charge-discharge cycles then.

The other cheap solution is a large Schottky diode from input to output. This results in the output voltage being slightly less than the input voltage. Again something that cheap makes may not care about.

See Teardown 1, or Teardown 2