There's a relation between King Turret and Portal 2 story?

Solution 1:

The Animal King Turret is first seen during Chapter 1 ("Courtesy Call") on the screen in the elevator room just before test chamber 2. As the player exits the elevator, the recorded announcer gives the following voice-over:

If the Earth is currently governed by a manner of animal-king, sentient cloud, or other governing body that either refuses to or is incapable of listening to reason, th--[RECORDING CUTS OUT]

Video source

The turret is just a comical representation of a ruler unwilling or incapable of listening to reason. There's nothing in-game which suggests that the above-ground has been taken over by a gigantic turret.