Higher octane is simply more resistant to detonation, which higher compression engines CAN have. Higher octane burns slower to resist knocking.
Most mower engines are 8:1 or maybe 8.5:1 compression. As in low compression, use any fuel you can buy and dump in. 87 octane is plenty in these engines.
Octane is for knocking. No power increase expected. This comes from the American thinking of more is better.
If all else fails, read your engine manual for proper octane guidance.
It is because Joe public really do not know the theory
Higher performance engines usually run higher compression, in fact a lot of older engines came with a choice of pistons from 7:1 through to 11:1 which is about the limit of petrol which is why we used to run methanol so we could go from 11:1 to 15:1
Methanol has less energy than petrol but the higher compression ratio liberates more power from the lower grade fuel than you can get from burning petrol.
However people see the more powerful versions of the engine requiring a higher octane so assume all of the power increase comes from the petrol & not from the higher compression ratio
Higher octane petrol usually does have a higher energy density than low octane fuel but not enough to account for the doubling of the HP from an engine that you can get from higher compression ratios.
When I had the time & money to play with things I was working on upping the compression ratio of some side valve engines
The 500cc engine I was playing with had a CR of 4.5 :1
We increased that to 7:1 which is getting near the limit for side valves and the HP went from 11 to 26
Based on he energy content increase of the fuel alone it would have gone from 11 to 11.25
FWIW we were aiming for 40Hp but the crankcase cracked and the con rod bent so it stopped right there
I think the K series HD's nearly doubled the output HP when you made ports in the head thus increasing the CR