One answer would be when Linnaeus, the father of modern taxonomy, classified whales as mammals in the 10th edition of his Systema Naturae, 1758. That link has a lot of info.

But really that wasn't a "realization" exactly--that is, back then it didn't mean that discovering the whale's anatomy meant a once-for-all understanding that they were mammals. Rather, there were different possible taxonomies, and where people classified whales depended on what they chose to emphasize.

So if you looked at the lack of legs and the fact that they lived in the water, a whale was a fish that happened to breathe air and bear live young. If you (like Linnaeus) looked at internal anatomy, it was a mammal that happened to look and act like a fish.

When people thought of species as separate, either taxonomy was just as plausible as the other; God could just as easily have thought "I'll make a fish with lungs" as "I'll make a mammal that swims." the debate went on well into the 19th century. Melville, in Moby-Dick (1851) took "the good old fashioned ground that the whale is a fish."

Then Darwin's theory of evolution (1859) created a new basis of taxonomy, based on the actual relatedness of species (this fit pretty well with Linnaean taxonomy, and today we use Linnaeus's terminology, but Linnaeus didn't think of species as changing or related by blood).

When people started thinking in evolutionary terms, only one taxonomical category made sense anymore--these were clearly mammals who had gone back to the sea, not fish who had independently developed lungs and mammaries.