Schnorr Identification Scheme under active adversary
It is important to consider the model in which security is proved. In this case, the attack proceeds in two phases: first a phase where the adversary as a verifier interacts with the honest prover, then a phase where the adversary as a prover interacts with an honest verifier. The adversary wins if the honest prover accepts. The paper proves, under reasonable assumptions, that the adversary cannot win in this model.

However (see the comment at the end of Section 2), this model is very restricted and not so useful. In more useful models where the adversary interacts with the honest prover and the honest verifier at the same time, the Schnorr identification protocol is absolutely hopeless, which isn't surprising since it wasn't designed with those models in mind.