The significant identity in "The Tell-Tale Heart" is the identity of the narrator. The entire story is told from his perspective. So, the two-part question is: how does his perspective form his own identity and how does this inform the reader of his identity?
First, speaking in terms of narration, the narrator is what we would call "unreliable." He doesn't know everything as a third-person omniscient narrator would and he lies (to himself, to the reader/auditor) and/or is simply delusional. Although the narrator tells the reader that he is reliable and even admirable, the reader knows otherwise. This is an odd kind of dramatic irony where the reader knows something the character does not: the fact that he is mentally unstable.
The story is presented in only one perspective, that of the narrator, but it does shift from logical to illogical. For example, he claims to have a "disease" (a symptom being that he is "dreadfully nervous") but adds that the disease has made him sharper. He also claims to love the old man: a rational thought. "He had never wronged me." But in his derangement, his irrational fear of the old man's eye overrides any love or sympathy he has for the old man. This is his derangement: that he can hold two completely different perspectives of the old man - love and fear.
The narrator addresses the reader directly in order to convince the reader (and maybe himself) that he is not crazy:
If you still think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body.
The narrator is so delusional that he tries to use his prowess at hiding his crime to prove that he is not mad. The invocation of the reader is not necessarily a shift in perspective in this case, because he always seems to be talking to someone (himself or the reader or an imaginary someone). But this shifting does illustrate how his unstable mind works.
When the police arrive, he tries to change his perspective/identity. Just as he tries to convince the reader he is not mad, he now tries to convince the police by taking on a new persona. He acts normal when they arrive and this seems to him to have worked. "My manner convinced them." But then he starts hearing the heart and loses it. His attempt to change his perspective/identity fails and he falls back into dreadful nervousness - leading to his confession. In the end, his unreliable perspective illustrates his mentally unstable identity, and the shifting underscores that instability.