A whale is no more a fish than a horse.

Is what correct? The grammar, or the zoology?

Actually they're both correct.

You've written two sentences, and they (potentially) mean different things.

"A whale is no more a fish than a horse." means "a whale is a fish to the same extent that a whale is a horse, that is, none."

"A whale is no more a fish than a horse is." means "A whale is a fish to the same extent that a horse is a fish, that is, none."

Suppose someone uses "the" instead of "a" before "horse", will it still be regarded as a correct sentence or an idiom?

Yes.