To add to Phil Frost's answer above, I suspect the 1.5kW number used in the US is a combination of practicality and convenience.
From a practicality standpoint, 1.5kW can be viewed as the point in the power to signal strength curve where your return on investment begins diminishing rapidly, at least in a hobbyist/residential scenario. To achieve a 1 S-unit improvement over a 1.5kW PEP signal, you'd need a whopping 6kW. At that point you begin to tax the typical infrastructure of a residential dwelling, as a Class AB amplifier would pull a bare minimum of 12kW to develop 6kW PEP (Class AB amps are very inefficient). In the US, the average home in the 1960s had wiring that could handle 15-20 amps, but not a heck of a lot more than that, so operating a large amplifier could pose a safety risk without special installation considerations. The grid was also not all that robust back then, so some guy constantly hammering on the grid with rapid 20kW on/off cycles is going to present technical problems for the utility company, not to mention the neighbors.
The necessary hardware to develop 6kW was also uncommon and very expensive back then, and even today would be cost prohibitive with modern components (enormous surplus Russian broadcast tubes excepted).
100-300 watts has been the "barefoot" standard for HF transmitters since at least the mid 1940s, primarily owing to the availability of affordable PA tubes in that range. If you're running 6kW or more, there will be such a disparity of power between you and the "average" ham (3+ S-units assuming identical antennas and symmetrical propagation) that you will be unable to hear many, perhaps most, of the people who can hear you. This increases the likelihood that you will unintentionally interfere with others. As your output power increases, the inevitable spurs and harmonics of your signal also increase in power, and filtering is not always a simple matter. With the current requirements for spectral purity (-50dB I believe), a 6kW amplifier could have spurs that approach 50mW PEP. On HF in good conditions, this is sufficient for reception hundreds of miles away. Achieving the -60dB or more that would be necessary to keep the HF spectrum fairly clean is not a trivial matter.
Finally I would posit that the precise number, 1,500 watts, was a bit of rounding for the sake of convenience. A 2 S-unit (12dB) increase above the typical 100w transmitter would be 1,600 watts. To make it a nice round number that is easy to remember, 1,500 was "close enough".