Catwoman has a long history in Gotham City as Batman’s classic anti-heroine femme fatale, but The Batman avoided using her iconic moniker and simply referred to her as Selina Kyle. In the comics and on screen, Selina has been a central figure in Bruce Wayne’s life. She has been a criminal to foil, a woman to fall for, and sometimes an ally to fight with. Her story has been told several different ways, but she always ends up with the nickname “Catwoman” as a high-stakes cat burglar with feline-themed attire. The portrayals of Selina Kyle in multiple TV series and big-screen appearances used that persona, but her most recent appearance in Matt Reeves’s The Batman fell short of actually using that name.
In The Batman, Selina Kyle (Zöe Kravitz) first appeared on screen as a server at the Iceberg Lounge where she donned a bit of a disguise in order to blend in better. Her first encounter with Batman was a brief scuffle after being caught breaking in to the mayor’s mansion. By the end of the story, Selina’s arc showed her become one of the city’s heroes as she helped foil Riddler’s plan to destroy Gotham. At her point of peak nobility, she chose to leave while Batman stayed to continue the fight as a symbol of hope. Unlike her costumed counterpart, she felt no higher purpose in Gotham. Consequently, her story in The Batman ended before she could gather enough of a reputation to receive the "Catwoman" moniker.
Selina’s character in The Batman was barely concerned with the world around her or making any kind of real difference in Gotham. The characters in the film with alter egos or nicknames were individuals who actually had a message to send. Bruce Wayne (Robert Pattinson) used the name “Batman” to represent vengeance and instill fear in the criminals of Gotham while Riddler (Paul Dano) intended for his name to inspire others to take down the corrupt elites of Gotham. With Penguin (Colin Farrell), it was evident his character had a reputation in the city, so the use of a nickname indicated his status in the criminal underground. Even “The Twins” (Charlie Carver and Max Carver) had their collective nickname to indicate that they're a well-known duo. With no need to inspire fear or respect in others, Selina had no need for a nickname.
Part of what makes Selina Kyle so iconic as Catwoman is her slick cat suit and feline-themed accessories. In The Batman, however, nothing about her indicated that theme other than two tiny spikes on the top of her ski mask. The only characters to use costumes were Batman and Riddler. Both Bruce Wayne's Batsuit and Riddler's costume in The Batman were useful because they conveyed a message while concealing the wearer's identity. They chose specific symbols to represent their messages and displayed it on what they wore. Selina concealed her identity with a mask of sorts and wore tight clothes for tactical and strategic reasons. She incorporated no symbol or theme into her outfit. Outfits as simple as hers would hardly warrant anointing her with a nickname.
The presence of a nickname is often an indication that its owner stands out from the crowd. Throughout The Batman, Selina spent most of her time hoping to remain unseen or unnoticed. Her priority was taking care of herself and the occasional stray, including her friend Annika Koslov (Hana Hrzic). Her efforts to obtain information from mob boss Carmine Falcone (John Turturro) in The Batman were more successful because of her ability to stay under the radar. Adopting a persona like Catwoman would only be appropriate if she meant to draw attention to herself.
As The Batman ended with Selina Kyle leaving Gotham to start a new life in Bludhaven, it seems likely her character will experience some kind of evolution if she pops up in a sequel or spinoff. It's quite possible audiences will see a story that eventually leads her to become the Catwoman figure with whom so many are familiar. Considering her story in The Batman, Selina Kyle was the only name she needed for now.