Water is formed from the reaction of an acid and a base. Why is it not classified as a salt?
Because a salt is formally the compound formed from the metal and non-metal counterions, when an acid and a base are neutralized to give water.
Now normally we represent a neutralization reaction as a net ionic equation, viz.:
Now, of course, these ionic species are not discrete entities; they are each accompanied by a counterion. The base is normally associated with a metal cation, and the acid by a non-metal anion.
Most of the time, the counterions, natrium and chloride, are just along for the ride; they are present in solution as
If I've missed the point of your question, I apologize.