Steve Pemberton stated in an interview (Mail February 25, 2017) that the only link between the stories is a small statue of a hare that he and Shearsmith place in each set.
As well as all of the episodes having a "bottle episode" aesthetic, several episodes also take place in "real-time" (the story takes place entirely within the thirty minute run time).
The title refers to the movie Gaslight (1944) starring Ingrid Bergman, where the murder and manipulation takes place at Number 9 Thornton Square.
Many television series include "bottle" episodes, where all of the action happens in one place or one set. The motivation is often to save money on one episode so more can be spent on others. However, the one linking aspect of this series is that all of the programs are "bottle" episodes, a single flat, or room, or a single piece of bedroom furniture. As a result, the original working title for the series was, apparently, "Message in a bottle".
The hare statue that appears in every episode was written into the storyline of the last episode of the fourth season.