There is an option in BIOS to auto-power on a PC, but even if it is configured to auto-power on, it should never experience the on-off loop. This is very unlikely a BIOS configuration issue.
It's possible but highly unlikely to be a power button issue. You didn't provide any details whether when the on-off loop happens, that unplugging and replugging actually successfully stop the loop. If the on-off loop still happens even after unplugging and replugging, you know it's not the power button. And that happened to my own PC.
It's probably not a dual-boot / software issue too. I used to have it on single-boot Windows system.
I don't have a particular answer but I'll try my best, if you follow through a particular experience I had, to lead to a likely problem with your mounting of the motherboard, and arrangement of hardware in the PC casing.
My experience
So on my older system, if I remember correctly, once the PC had shut down, after a few seconds the PC would automatically turn on (i.e. the fans would spin up and the motherboard LEDs, power LEDs will turn on) for about 5 seconds and then the system would completely turn off (fans and LEDs off). Like you said, the turning on and turning off would repeat endlessly.
I don't remember exactly solving the issue, but whenever the auto-turn-on happened, I'll hold down the power button for several seconds, which actually forces the system's power supply to shut down by most BIOS'es (unless specifically configured not to). This sometimes successfully made it stop auto-turning-on; but when it failed I would have to turn off the PSU's switch or the plug's switch, then hold down the power for a several seconds - in order to drain out the capacitors of any remaining charge. Then turn the PSU's / plug's switch on again and that would sometimes fix the problem; but it was when it didn't that actually provides a clue to the cause: Turning the PSU / plug switch back on, the PC turns itself on automatically, going back into the on-off loop.
If your issue is not the same up to this point, then forget about my answer. If not, continue reading.
The fact that we drained the motherboard's capacitors of charge when the mains was off should reset them, so that when the PSU / main's switch is turned on, the PC should not turn on itself automatically; But does in on-off loop. Which leads us to some shorting happening in the PC case.
Possible fixes that worked for me
The first thing I would do, is to ensure that the motherboard's mounting screws are not too tight. If you have clunky hardware (like big GPUs, CPU heatsinks), it can cause the motherboard to bend ever so slightly so that part of the motherboard touches the casing. That contact between the motherboard and the case can lead to a electric short which can trigger a pc go into that crazy loop.
The second thing is to check to make sure all hardware components are not touching each other excessively. Keep the inside of the casing tidy.
When I did so, the frequency of this on-off loop drastically fell. Although I didn't completely solve the problem, it mostly fixed it for me, for some time before it began to happen again more often. Over time the components in the PC shift slightly from minor knocks or moving of the PC casing. So it would be great to have a good casing that would not be so susceptible to such issues.