If I introduce a problem, say as an ILP formulation, should I also discuss the number of introduced constraints? If yes, why?
In academic publications (where the point is to present a model and possibly computational scheme to solve an actual problem) I typically do not bother to count the constraints. First, the reader can do it themselves from the algebraic formulation. Second, the reader is more interested in whether "realistic" instances of problem are solvable with "reasonable" hardware in "reasonable" time, which they find out in the experimental results.
I do agree with the comment that formulations requiring an exponential number of constraints are potentially problematic, if there is a need for them to scale. I say "if" because in some cases you are presenting computational results for problems of the size that users will likely encounter. For instance, if you are creating bus routes for school children, the fact that computation time blows up as the number of kids attending the school tends to infinity is probably not very relevant.
I agree with @prubin on academic publications.
However if your audience consist of "laymen", e.g. when working as a consultant, a discussion can be helpful: Providing insight into how a change of parameters affects the number of constraints (and more importantly: the number of decisions to be made, i.e. the number of variables) helps them to get a better understanding of the complexity of the underlying problem.
Reconsidering @prubin's example: One could present a table showing the number of constraints (and variables) for 5, 10, 20, .... school children.