You might think that your phone is directly charged from the powerbank/charger, well it is not.
The powerbank or charger only have the job to provide 5 V up to
the current stated on the device. So 5 V, 1A means a load (like a phone that's charging) can draw up to 1 A. The voltage is and must be 5 V. The phone determines how much current it chooses to draw.
How much current the phone will draw depends on signals on the 5V USB outputs of the powerbank or charger. If the powerbank or charger does not provide the correct signal the phone will "play it safe" and only draw 0.5 A or 1 A depending on the model.
The phone has the actual battery charging circuit, it MUST be in the phone as it needs to "know" what voltage the battery must be charged to, how much charge it has and what temperature the battery has.
Also the phone charging slowly when the battery is 80% full is actually a good thing and should also happen when the phone is charged from the 1A socket. Lithium based batteries MUST be charged more slowly when they're almost full. If this is not done the battery will wear out more quickly.
Also when the battery as at a very low charge, the charging current must be kept lower. Only at around 30% - 70 % charge is it OK to charge with the maximum current. The phone's build in charging circuit takes care of this so it happens automatically.