The comparison is not really correct, because the "hp" was defined as the power that a horse could sustain for a very long time, like when being used to power a pump or a mill, which was the right equivalence for the replacement of a horse with a steam engine.

Like humans or any other animals, horses can sustain much higher powers for short times, than for long times. Few horses, if any, could sustain a power of "1 hp" for hours.

Seems more likely marketing got involved.

It's correct for their audience's curiosity, comparing the peak horsepower of a sports car to a horse.

Then you could simply attach your 1hp engine to some simple mechanical device (e.g., an elevated water tank) that will deliver any amount of horsepower after having been "charged" by your engine for some time, and call that "peak power". I.e., the peak power measure doesn't make sense: the only number that counts is the average, sustained output.

But that's less fun!

It is important in practice. Most machines need short burst of peak power when starting.