Sort of. Roughly speaking, elev is used for primary-secondary education, student for late secondary education and thereafter.
Is it? I don't know very much about the French educational system, but it sounds reasonable.
Hej! In France we say "élève" as long as you are in "l'école" and in "le collège et le lycée", so from three to eighteen years old. After that, you go, for example to university and are called " un étudiant "or "une étudiante". :-)
I suppose, you kind of use them in the same situations, although a teacher at university could just as easily refer to their students as "mes élèves", the students themselves wouldn't refer to themselves that way.
It is similar in German - "ein Student" is at university, but "ein Schüler" is still in grade school.