(Disclaimer: this is not my field.)
Let g be the chosen generator and n its (prime) order. In the ECDSA algorithm, these are publicly known. It is true that g generates a cyclic (abelian) group isomorphic to Z/nZ.
Now a private key consists of an integer k, and the corresponding public key is the group element h = kg. (I use additive notation since we are in an abelian group, so kg = g+g+...+g (k times)). If we were actually working in Z/nZ, it would be trivial to recover k from h: just divide h by g (mod n) using the Euclidean algorithm. Then we would have no security at all.
The point, as I understand it, is that in the elliptic curve group, there is no known efficient way to "divide". And while there certainly exist isomorphisms from
<g> to Z/nZ (map g to any element of Z/nZ you like), the inverse of such an isomorphism is not trivial to compute.