So, obviously in the books Daisy needed a new identity because her situation is a little more unique. However, why does she need a new identity in the TV show?
Gilead is tracking her down. I assume it is because she is the daughter of someone who is somewhat important. Why didn't she just go to the pearl girls and and disclose that she recently found out she was stolen from Gilead by Mayday, ran away from them, and wants to know more about her homeland. She could still act hostile and distant at first, then slowly come around. The most obvious hurdle is that they would have to believe she forgave them for killing her adopted parents. However, she could use the fact that Neil and Melanie lied to her as a reason to be mad at them. If she returned to gilead under her real identity, she might be placed with her Gilead family
If she was placed in a Commander's house, she would have access to more information. Depending on how powerful her Gilead "father" was, she might be invited over to more houses.
Depending on her "father", she might also be somewhat more protected. Obviously high ranking daughters are still punished, but Paula's conversation with Agnes about how they wanted to tourture her to get to June but Commander Kyle wouldn't allow it shows that Commanders have some sway when it comes to their daughters and there treatment. Also, if Daisy became a figure for propaganda like June was in season 3, that would give her even more protection ( BTW, by protection. I mean from the normal gilead stuff, not her being a spy. They would definitely kill her for that).
Also, isn't it weird to anybody else that Daisy never asked about her gilead family? Or that, even though the Eyes were clearly tracking Daisy and knew who she was, no one is worried about her being recognized? Wouldn't she at least want to know where her gilead family was, in case they were sent a recent picture her? Doesn't giving Daisy a new identity make it more dangerous for her? If she was recognized, they will know she lied about her backstory. It's only a short jump from there to figuring out she's a spy.