Theoretically, 'rated' power' is the power the motor/controller can handle continuously that will cause it to reach thermal equilibrium at a temp lower than will incur damage. Thermal equilibrium occurs where: (heat from power lost due to device efficiency) = (heat dissipated).
This is like water = heat, power is pouring water into a bucket, and heat dissipation is a big leak in the bucket. Up until some point, you can keep pouring water in faster and faster and the bucket will never fill up. At some point, the 'rate in' exceeds the 'rate out' and either slowly or rapidly the bucket will fill up, spill over, and your motor will cook. Similalry, if you back off on the pouring, it takes a bit for the water level to go down (heat to dissipate).
So - a 600W motor should run all day at 600W, but realistically can take thousands of watts for a short burst - but don't 'burst' again repeatedly before the heat bleeds off.
Sadly, motor manufacturers are notorious for bogus ratings. See the ebikes.ca motor
and (new) trip
simulators to see heat estimates made based on real motor dyno tests.