Any time you are using physics engine in game and need to control a thing while maintaining proper physics interactions a PID controller might be used. You could achieve desired state by cheating and directly animating position or velocity (which gamedevs often do), but if you want to maintain proper interactions with other physics objects the best way is to control force. How do you calculate required force - using a PID controller.
It could be a moving enemy, flying robot or a spaceship, rotating gun turret, maybe even doors which need to be moved to desired position.
Even without physics engines PID controllers can be useful in games for making procedural animations. Having an object suddenly starting and stopping a movement doesn't look natural. It looks much nicer if initially an object slightly lags behind, slows down closer to target and maybe even slightly overshoots the target. It looks more natural because that's how things move in real life even non mechanical things. Try quickly swing a hand and then stopping it, you will see a little bit of overshoot/oscillation at the end.
PID controller isn't the only solution to the situations described above and sometimes it's an overkill. Sometimes you might just script or animate certain behavior. Sometimes you might use a simplified control model. Sometimes you might implement P, and intentionally or not achieve other components by adjusting physics object properties like friction (unfortunately you can't easily change physics properties in real world). With regards to animation example similar look can be achieved by directly animating it and any good animator will be familiar with things like ease in, ease out and overshoot, that's fine for static animations but games are interactive so it's nice if things react to player movements. Procedural animations can also be achieved by simply using the physics engine and letting things dangle, but that will probably be more computationally expensive than simple PID controller, also harder to constrain and prevent violent shaking in case of unexpected collisions with other world objects especially when some of them are manually animated.
Each approach has their own tradeoffs. And which one gets chosen will depend on the specific situation and skillset of people making the game.
There are couple of factors that make using a PID controller in games a bit easier compared to real life. In games you can usually directly read the exact positions and speeds of all objects, you can generate force out of thin air, you can generate unrealistic forces and if something goes wrong you can simply clamp the numbers. In games you also choose at which level to operate the PID loop thus making tradeof of code complexity and while still getting benefits of nicer looking animations. In real world you might want to control position of flying quadcopter while the thing you can directly control is speed of propellers with linear velocity, linear acceleration, angle, angular acceleration being between without a way to skip them.