Rating a motor at "1000 watts" simply means that it will take that amount of power without melting. For HOW LONG is a question which is answered by advertisers, not engineers. Often the identical motor will be rated at 1000, 1500, and 2000 watts, just not for the same amount of time. Check the weight. Sustaining more power generally requires more physical mass, in similar designs. More pounds of iron and copper will soak up more heat. Put an oil fill and external heat sinks on a "1000 watt motor", and you can safely run 3000 watts for short periods. More if you live in a cold climate.
For a battery, it is a much more accurate measurement, but something of a moving target. Batteries degrade over time and loose capacity. Batteries can supply a large amount of amps, for a short period, but usually this will degrade them faster. Drawing a lower amount of amps will result in a battery with a longer lifetime, also greater useful capacity for an individual charge cycle.
Think of jogging for a long distance, or sprinting for a short distance. Then do both when you are much older. What is your capacity in these different situations? How does each different usage condition affect your total output?