I think you must think about effective radiated power. A directional antenna concentrates the power in one direction so it seems to have much more power than an Omni-directional antenna. Its effective radiated power is much higher than its total power.
Some antennas spray the sky and ground with wasted power. They also spray their power all over the place. But good antennas direct their power to where it is wanted.
Look at space craft. They have very low total power but the very high directionality of their antenna and of their receiving antenna allow them to have a very high effective power.