What is the purpose of classifying fighters?
People like to classify things, to divide them into categories.  Categorizing allows people to get an initial broad understanding of a complicated subject more quickly.  It is therefore very useful for purposes such as marketing and articles aimed for people who aren't necessarily subject matter experts.  However, classification is necessarily a crude tool that often isn't exact, leading to confusion and arguments about where the category boundaries lie.

People who are subject matter experts, such as designers of a new fighter plane, or military analysts trying to determine the capabilities of a new secret aircraft produced by other countries, presumably don't spend much time wondering into which category to put a given aircraft.  Instead they focus their time on the details of the subject, the same details that are too much information for the person trying to get the quick grasp of the complicated subject.

To answer your questions specifically, knowing the generation of a fighter gives someone a rough idea of its capabilities, and allows the person to quickly compare a given fighter aircraft to other fighter aircraft.  Knowing the generation of a fighter is useful for people who aren't experts on the subject who need a quick broad understanding of the capabilities of historic and modern fighter aircraft, and who need to know roughly where a given aircraft fits into the large picture.