Identity is a slippery thing in The Island of Dr. Moreau, and at no point is it clear cut. From the outset, Prendick has a difficult time telling the Beast Folk apart from humans. Even when the physical distinctions become clear, the internal distinctions get all muddy. Prendick begins to see bits and pieces of humanity in the fears and plights of the Beast Folk. Meanwhile, he sees beastliness in the actions and offenses of his human brothers. By the end of the novel, Prendick has no idea where one stops and the other begins. But that's kind of refreshing, right? In real life, identity is complicated and unpredictable. Why shouldn't it be the same in our stories?
Prendick's tale is really one about keeping his identity. As he delves deeper and deeper into the island, he beings to lose his sense of self to the island, culminating in his own animal reversion. Ultimately, it's left ambiguous as to whether or not he succeeds in remaining himself.
On the other hand, Montgomery's story is about losing his identity. His talks with Prendick about London are his last grasping attempts to hold onto his old civilized self. With every day, every drink, on the island, Montgomery loses more of himself to the place.