The phrase "When you give them an inch, they will take a mile" reflects a form of flawed reasoning known as false dichotomy. This reasoning suggests that there are only two outcomes: either allowing some leniency will lead to excessive demands or it won't. It simplifies a complex situation into an exaggerated choice, ignoring possible moderate outcomes.
However, it might also be argued that it appeals to emotion as it plays on fears of being taken advantage of.
If I had to choose one based on common interpretations, I would lean towards false dichotomy as the most appropriate label for the reasoning involved in that phrase.