The Batmobile has undergone significant changes over the years, but the car has remained an integral part of Batman's time as a superhero. The answer might seem obvious, but why is his car called the Batmobile? Because, that's what Batman wanted to call it, god damn it.
The Batmobile debuted alongside the Caped Crusader in Detective Comics #27. Over the years it has evolved from a simple car, to a bat-themed black vehicle, to a technologically advanced armored car. The Batmobile's look and features has ranged from fun and silly to tactical and intimidating. No matter how Batman's trusted vehicle has evolved, it's always been called the Batmobile. But, don't question the name in front of Batman.
All-Star Batman & Robin, The Boy Wonder is one of the most divisive comics ever made, as the series depiction of Batman is tonally inconsistent with how the character has always been portrayed. In the series, he's violent, rude, and mean. While Batman remains a hero, he's not likable at all. On top of his mischaracterization, Batman had some groanworthy one-liners. After being up on thugs and having sex with Black Canary (with their costumes on), Batman offers her a ride home in his Batmobile. "Batmobile?" Canary asks. He responds in a tirade, telling her "not one word. I've taken enough grief about calling my goddamn car the goddamn Batmobile."
Batman than offers a reason why he called his car the Batmobile, telling Canary "I'm the goddamn Batman and I can call my goddamn car whatever the hell I want to call it." Canary tells him "whatever you say, man of mine. That's just a totally queer name for a car, is all." The dialogue is cringeworthy, but it does give a reason to why Batman called his car the Batmobile: as Miller loved to write throughout the series, he can because he's the "goddamn Batman."
Batman really wanted Canary to know he could call the Batmobile whatever he wanted. But, he chose Batmobile because it made the most sense and because he could. It's sort of a painful explanation - as all that's really needed for him to say is, "I'm Batman, and I like naming things after bats." Instead, he comes across like a major jerk who is set on doing things his own way, because he's the "goddamn Batman."