The French "Grandes Ecoles" system of higher education is very difficult to understand for foreigners. Pointing out the features that are unique to that system only creates perplexity and confusion. Instead, let me point out some features that are common with some schools in the US (the other system with which I am familiar).
Consider the "Ecole Normale Supérieure de la rue d'Ulm", quite a mouthful of a name! (It is also called "Ecole Normale Supérieure", "ENS", "ENS Ulm", "rue d'Ulm", or "Ulm" for short.)
So I think that someone from the US and who would be looking for a rough and simple analogy could consider that ENS Ulm, ENS Lyon and ENS Cachan are the French Ivy League schools, and, among those, consider that ENS is the French Harvard. (Similarly one might perhaps consider that Ecole Polytechnique is analogous to CalTech.)
The single most outstanding difference between ENS and Harvard lies in their mission: Harvard educates students to serve society, but the particular ways in which they will serve are left undefined. In contrast, the three ENS schools (Ulm, Cachan, Lyon) exist to educate students who will serve society in well-defined careers: the students are primarily future academics, teachers, and researchers. (Thus, for example, Ivy League schools recruit students with excellent grades but who also additionally show some other, non-academic potential, whereas the ENS schools recruit solely on the basis of academic performance.)
The next most important difference is financial. The ENS are public, and the French state aims to offer the best possible conditions to help its most promising young people become leading academics: one way in which it does that is by paying for their studies for 4 years (in return, the students commit to serve the state for 10 years.) That is, the ENS pays the students, whereas Harvard asks the students to pay.