For a while motors were sold with frankly false claims, using stall current draw to describe 'peak horsepower'. A marketing person's wet dream. In the 1950s motors were rated in horsepower by checking sustained ft-lbs/sec which is more accurate. In those days motors were far less efficient (between 70 and 80%) but I still use my Dad's old repulsion motor from his table saw, made in the 1930s, and it still works well.
For air compressors, someone invents some kind of criteria for putting a bigger number than everyone else on their ads. I only check the current draw/voltage on a compressor and totally ignore the rest of it. Some of these guys are paid to mislead, and they do it very well. Even the output air claims can be grossly inflated.
From an engineering perspective, If you are putting 746 sustained watts into a motor with a piston compressor, the mechanical conversion factor is something like 40% so a 1 HP motor can only produce a little less than a half that in work using the output air. I am very reluctant to describe any compressor output in HP. FWIW, the silent compressors that use double pistons and 'oiless' compressors are less efficient at converting motor energy, but I couldn't find examples of their conversion efficiency easily.
I could give a serious rant about vacuums, but it would be less civil