It is possible for something to have failed in the jump start circuitry that gives you 12V and high current (vs the 5V that charges a phone, or that recharges the battery). And the battery itself would still be functional to turn on lights and communicate until it is under a heavy load.
It is also normal for the blue lights to go off after a period of time. I have never watched them while trying to jump start the car though, I am usually behind the steering wheel at that point.
I would try a few things:
Will it charge a phone over the USB port? - If yes, the battery is still working on some level, if no it is just not holding a charge.
Check the voltage (DC, should be 12V or less) on your battery. When it is low (down to at least 11V, ideally lower for a good test; a fully charged battery should be 12V), hook up the jump starter, make sure it is on, and check that voltage again. If it has gone up noticeably, the jump starter is providing power to the battery. Wait a few minutes, check it again and see if it has risen further - if it tops out at something below 10V, then it is working but not able to provide enough power to jump start the car, and is effectively dead as a jump starter.
Unfortunately, Aukey seems to have discontinued making car jump starters. It is a limited market space I think. And this model was discontinued over 2 years ago I believe, so yours is at least that old - I got mine in 2016.