Go into BIOS and disable Power On Features like WoL or others that dont use the power button, then save the changes. You might have WoL enabled. Also seen a keyboard do this too with a cat around and a key gets bumped with Power on feature of by key press.
https://en.wikipedia.org/wiki/Wake-on-LANIf its not this as the cause then it could also be a faulty power supply with ripple, or a motherboard with a capacitor that has failed that is allowing for ripple to trigger a false soft-power on condition. The power button when pressed uses the soft-power on state. If the motherboard thinks that the button was tapped when it wasnt it can boot it back up.
Hopefully its an easy BIOS config change with WoL. I had a Zotac motherboard system come with WoL enabled by default and for some reason a WoL instruction was on my local network causing it to boot up on its own, not sure which device was causing WoL to sense a boot condition either. Disabled WoL and problem went away.