In-universe: this is unexplained, but sort of makes sense.
Kara Zor-El arrives on Earth as 14 year-old girl. She's a newcomer on a strange planet, and is about to have a hell of a time learning to fit in. One of the ways to make this a touch easier for her would be something as simple as continuing to use her real first name, which, thankfully is a not-uncommon name on Earth as well.
Out-of-universe: this is unexplained, but sort of makes sense.
While it's true that the TV show Supergirl slightly pre-dates the publication of the comic Supergirl Rebirth (though which one entered early production first is unclear), it is not at all uncommon for writers of cinematic versions and comic versions of characters to get on the same page. In the rebooted Supergirl Rebirth, Kara Zor-El lives under the assumed name of Kara Danvers.
By keeping the same name between versions, people picking up comics after watching the TV show have an easier transition into the comic book world.