Very good question. When thinking about the meaning of sense of there is a difference in meaning between sense and sensation. It is true that sense can mean feeling, like a sense of tiredness or a sense of hunger.
However it can also, when in the context of an ideal or a virtue or an ability (such as humor or justice or irony or direction or responsibility), mean the ability to appreciate or understand something. This is how it is being used here. So she is saying that the customers in these restaurants feel or understand the idea of urgency, of not having a lot of time.
Ex: My husband has a great sense of humour.
Ex: I have a terrible sense of direction. I'm always getting lost.