What is a monotone dataset and monotone classification?
A monotone classifier is a classifier that is a monotonic function.

One reasonable definition of monotone data set is that it is a data set that is consistent with some monotone model (i.e., where there exists a monotone classifier that gets 100% accuracy).  I don't know if this is a standard definition so hopefully any text that uses that phrase would provide a definition.

Why use a monotone classifier in the setting you mention? Perhaps we have some a priori reason for believing that, all else being equal, the higher the students rate their satisfaction, the higher the quality of the course.  That amounts to a monotonicity assumption about the underlying model that relates quality to satisfaction.  If we assume that the data was generated from a monotone model, it is reasonable to try to learn a monotone classifier.  Whether this is appropriate in any particular setting depends on whether you have a reason to think that the data does come from a monotone process/model or not.