Troubleshooting why Powerbank Doesn't Charge Cell Phone
A device (phone) gets charged at optimal (high) rate only when it recognizes "charging port signature" on charger side. For Type-A port, there are several DIFFERENT port signatures that a charger port can provide:


USB Battery Charging v1.2 Dedicated Charging Port, where D+ and D- are floating but tied together;
USB Battery Charging v1.2 Charging Downstream Port, when a special low-voltage handshake over D+ and D- lines happens;
Apple charger signature, when a combination of DC levels is present on D+ and D- lines with high-impedance resistor dividers (75k/50k);
Sony charger signature, when D+ and D- have 5k/10k dividers to VBUS;
Qualcomm "Quick Charger" signature, which uses an intelligent IC on charger side to understand a special sequence of DC levels driven by device on D+ and D-; can provide 9V and 12V on VBUS;
Samsung charger signature (details unknown, to me);
USB Power Delivery protocol V1.2 - uses complex packet-based protocol over VBUS wire using FSK modulation method at 23.2MHz carrier frequency. The port might have additional contacts to detect plug insertion.
...


For Type-C connector, there is a newer Power Delivery protocol v2.1, which solely uses the CC wire (Communication Channel) to negotiate voltages and available currents that can be delivered via VBUS.

A charging port usually provides only one signature, and most of mobile devices have an intelligence to recognize it, usually by sequentially probing different signatures. When a phone recognizes something it understands, it will take the charge. [However, there were attempts to make a charging port that tries several signatures sequentially, as some Microchip/SMSC hubs of the past.] 

We don't know what kind of signature is used in this particular power bank, so all bets are off. Now, consider that your old phone can be designed for some other proprietary signature, so it should not come at any surprise that some phones will not charge well (or at all) from this particular power bank.