Hopefully things start to appear much clearer now.
Indeed the two devices that I bought, the power charger and the battery pack, both implement Qualcomm QC 3.0 technology according to which the delivered higher power is managed via a “negotiated” higher voltage instead of a higher current, in this way avoiding wire overheating and possible detrimental effects on the batteries under charge.
The method is called Dual Charge (or Parallel Charging) using two PMCIs (Power management integrated circuit) to split the power into two separate streams.
Maybe supplying 9V/2A as 4.5V+4.5V to the two Li-ion power pack batteries. Is it something like that?
This makes it sound very complicated, but typically a power bank would be multiple 3.7V 18650 cells connected in parallel.
This is a cheap QC3 board:
The PMIC is obviously a TI unithttps://www.ti.com/product/TPS61088
You put 3.7V in and you step it up to any variable number of volts between 5 and 12V.
Clearly the issue with a 20 Ah power bank is that you have six cells typically, so as they age it might be better to charge them separately; in terms of charging the cells then essentially you put 4.2V or less into them, and they charge but with age some cells might get damaged so if for example you put 4.2V into six cells at once, then that's not as good for the cells as if you charged all six separately.
Your powerbank is claiming to have two LiPo cells, instead of six Li-Ion cells, which is unusual, but if they are in fact charged independently that would be a good thing (they are still 3.7V btw, not 4.5V).
However none of this (how the power bank is charged) has anything to do with your output spec, which relates to the circuit board at the top, and where for instance if you want to charge a phone at say 24W, then 10V x 2.4A, or 12V x 2A is much better than 5V * 4.8A, since the resistance/heat in a wire relates only to current, not voltage, so where the wattage over a given cable/conductor has increased over time, it makes sense to increase voltage, which can only be done following the IC negotiating a safe higher voltage with another IC in the device being charged, failing which it will default to 5V.