We learn Clarice Starling's job before we learn her name. She's wearing a sweatshirt that has "FBI Academy" embroidered on it. That is the critical part of her identity that matters. Her goal is to become a full-fledged FBI agent.
"Bill Skins Fifth"
Here is something else we learn from text on the screen. When Starling enters Crawford's office, he has newspaper articles all over the wall. This shows us the identity of the antagonist, and what his motivations are: skinning people.
"Hannibal the Cannibal."
This is one more bit of exposition telling us the identity of our third major player: Hannibal Lecter. Clarice knows him as "Hannibal the Cannibal." His reputation precedes him.
CHILTON: Oh, he's a monster. A pure psychopath. So rare to capture one alive. From a research point of view, Lecter is our most prized asset.
Dr. Chilton up-sells Hannibal Lecter's mystique. At this point, we've heard him called "monster," "cannibal," and "psychopath" before we see him. All these adjectives shape our image of him.
LECTER: You're not more than one generation from poor, white trash, are you, Agent Starling?
Dr. Lecter is a skilled therapist, able to analyze Clarice's background just from talking to her. However, he uses this insight against her. He sees her trying to hide that part of her identity, and he pulls it out to insult her.
CLARICE: If he sees Catherine as a person and not just an object, it's harder to tear her up.
Senator Martin wants people, especially Buffalo Bill, to see her daughter, Catherine, as her daughter Catherine, not as an anonymous victim or an object. Establishing her identity is important for humanizing her. So in her television plea, she uses her name over and over.
LECTER: Billy hates his own identity, you see, and he thinks that makes him a transsexual. But his pathology is a thousand times more savage and more terrifying.
Buffalo Bill will try anything to try on a different identity because he hates himself too much. If he weren't caught at the end, he would finish his woman suit and probably still hate himself. Then he'd probably start skinning actual lambs to be a human sheep.
MURRAY: Is it true what they're saying? […] He's some kind of vampire?
CLARICE: They don't have a name for what he is.
There's a mystique around Hannibal Lecter, like he's some sort of mythical figure or super-villain. We have a feeling that Dr. Chilton might be spreading some of these rumors. The scarier Dr. Lecter appears, the more heroic Chilton seems for keeping him behind bars.