A zero-knowledge proof is a protocol by which the Prover demonstrate to the Verifier that he knows the solution to a given problem, without giving to the Verifier any additional information about the solution -- that is, no information that the Verifier could not already obtain alone. In the case of the discrete logarithm, the y value is not part of what the Prover sends; it is a given. The base problem consists in (p,g,y) and what the Prover tries to demonstrate is that he knows the corresponding x -- but without helping the Verifier find out x.
The Verifier can always try to solve the problem by himself. This is in fact unavoidable: since a ZK proof is about proving knowledge of a solution to a given problem, anybody can, conceptually, try to solve the problem upfront, if only by exhaustive search of all possible bit patterns for a solution.
This does not count as a leak because it is not information obtained from the Prover. It is part of the context. It is information that everybody already has. The whole of ZK is about demonstrating knowledge of some value without yielding extra information that was not already public knowledge.
(Honestly I never understood why the cave example was all the rage. It is an analogy and it breaks down when looked at too closely. For instance, Victor could say to Peggy: enter the left corridor, and exit from the right one. This would demonstrate that Peggy knows how to open the door, and would be vastly simpler than the pseudo-ZK concocted for that example.)