You've correctly shown that maximizing the likelihood is equivalent to minimizing cross entropy in the discrete case, but frankly that is unrelated to your clai

You've correctly shown that maximizing the likelihood is equivalent to minimizing cross entropy in the discrete case, but frankly that is unrelated to your claim that the equivalency doesn't hold in the general case. As noted in the sibling comment, the generalization to the continuous case is evident when viewing the empirical data distribution as a mixture of dirac densities.