Neither the Golem nor the Jinni start out with a name, hence them being called the Golem and the Jinni for most of the book, unless they're being referred to by humans who only know them as Chava and Ahmad. These names are assigned to them, though, and they never quite identity with them.
It was not so much that he liked [the name "Ahmad"], as that he found it the least objectionable. In the repeated a's he heard the sound of the wind, the distant echo of his former life. (5.48)
The Jinni will never fully accepts an American identity, but he makes a small concession in accepting a name.
After that, there was little question as to his path. […] Doctor Mahmoud became Ice Cream Saleh. (5.86)
The supernatural creatures aren't the only ones to have identity changes. Most of the characters end up altering identities when coming to America, and sometimes it's a bit of a downgrade, like going from doctor to ice cream seller.
To leave everyone she had ever known, and live with a strange man, and lie beneath him, and be ordered about by his family—was it not like dying, in a way? Certainly she wouldn't be Fadwa al-Hadid anymore. (5.123)
Fadwa's ruminations on marriage make us wonder just how much a person's identity is compromised when getting married. How have things changed since these days, when arranged marriage was common?
"I know [I need a name]." She smiled. "But I'd like you to choose it for me." (6.35)
The Golem, being used to having a master, doesn't even want to define her own identity at first. She wants someone else to do it for her. Good thing the Rabbi has good taste and picks Chava instead of something terrible.
His attention turned from the mirror to his own face reflected in it. He'd seen it before, of course, but never so clearly. (9.28)
The Jinni identifies more as a being than as a person, though for us, it's easy to forget that this human form is one he's trapped in. Being in a human shape ends up shaping his identity—it would be a totally different book if he were shaped like a coyote or a cactus or a chicken nugget.
Although [the Jinni] might be forced to live like a human, he'd never truly be one. (16.275)
This is a good elaboration of the previous quote. The Jinni has to let his form dictate his behavior. And by paying attention to the Syrian Desert flashbacks, you can see that the humans never quite make a lot of sense to him. It's hard to be something you don't truly understand; he'll always be a little different.
"If you could do whatever you wanted, without worrying about staying hidden? Would you still work at a bakery?" (17.45)
The Golem is only about six months old, so this is the Jinni's way of asking her what she wants to be when she grows up. What would a clay creature of Jewish mysticism really want to be? A rabbi? A Pottery Barn manager?
[The Golem had] changed just enough to wonder if she was still herself. (18.57)
The Jinni starts bringing out the Golem's more reckless side—running through the streets of New York, going out for walks at night—and she starts questioning her identity. What is her true identity? Is it how she acts, or how she was made?
As he dressed quietly and found his shoes, the role of Joseph Schall fell from him like a skin. It was near midnight, and Yehudah Schaalman's day was just beginning. (22.1)
Schall/Schaalman is yet another dude with multiple identities, and his names clue us in to what he's doing—helping at the Sheltering House or casting black magic—with Schall being the good side, and Schaalman the evil.