powering old portable 80's radio with powerbank
I suspect that your radio is drawing too little current making the power bank think that there is no load and switching off the 5 V.

Possible solutions:

Use a different power bank that does not switch off

Draw more current, for example by connecting a 6V light bulb (bicycle light) in parallel with the 5 V.

Draw more current by switching on the cassette recorder function, you can try that now and see if it prevents switching off.