So, Crowley goes strictly by a single name for millennia, right? But sometime prior to his meeting with Aziraphale in the church in 1941, he's started going by Anthony J. Crowley. Why is that?
I've got a theory/headcanon.
The fandom generally agrees that Crowley fell in love with Aziraphale first. He's been smitten for a long time, and every time he and Aziraphale fight, he's the one who's the most wounded by it. Yet he also still clearly loves Aziraphale and keeps finding his way back to the Angel in the end.
Now, imagine the pair have had a fight, either the one in the park over the holy water, or one after. Crowley is hurt and sulking, and goes back home to fill out the latest round of paperwork for the Head Office.
He fills out the report on whatever blasphemous deeds he's been doing while brooding and grumbling about the hard-headed angel that wouldn't know flirting if it literally grabbed him by the wings and pulled. He's making himself more and more upset by the minute.
Crowley's so upset, in fact, that he doesn't realize he's been doodling in the margins of his reports. Doodling things like "A+C", though thankfully not using hearts. But Crowley doesn't realize that he's doodled until AFTER he turns the report in and Beelzebub calls him in for a meeting.
"Crowley, would you care to exzzplain yourself?"
The Lord of the Flies points to the doodles and Crowley feels the entire weight of the universe settle somewhere in the pit of his stomach. He sees them torturing him, Heaven killing Aziraphale, everything turning to absolute shit. This is bad, and he knows it.
But his mouth talks before his brain can catch up, and he says, "It's my initials."
Beelzebub stares at him. "What?"
Crowley charges on. "Initials. Human stuff. If you just go by one name they think you're a bit, well, thick. I've been trying out a new alias to blend in better."
He doesn't stop to think that perhaps Beelzebub, no other names given, might resent being called "a bit thick."
Fortunately, she doesn't seem to pick up on the implied insult. "What doeszzz the A szzztand for?" Beelzebub demands.
"Anthony." Crowley hates it immediately, but it was the first name beginning with an A that wasn't Adam he could think of. Just why he thought of Anthony he never could reason out.
"And the T?" Beelzebub inquires further.
Crowley's mind has gone completely blank. He cannot think of a single name beginning with T. "Not a T," he says instead. "It's a J." He makes a mental note to kick himself later, if he survives this encounter.
Beelzebub squints at the letter. "It'szzz a J," she repeats. "Really."
"Yep. It's a J. Sorry, it's just that my handwriting is terrible."
The other Demon doesn't so much as crack a smile. "What doeszzz the J stand for, then?"
And of course, Crowley can't think of a single name that starts with J either. So he says, "It's just a J really. The first name is really the important bit. Everything else is just for show."
And Crowley stands there, wondering just how much holy water they're going to dissolve him in, for what feels like much longer than the nearly 6000 years he's spent on Earth, before Beelzebub finally rolls her eyes and dismisses him.
"Keep the reportzzzs neater," she scolds. "Markszzz in the marginszzz are a szzzign of szzzloppy workmanszzzhip."
"Right. Of course. Will do." Crowley manages not to run out of the office, but barely.
So of course, to keep his cover story intact, Crowley has to make good on his statement. He styles himself as Anthony J. Crowley, signing documents and giving his name to everyone who asks for it. He is obsessive to the point of carelessness, using his name openly when before he would have been more cautious. Soon, everyone knows who Anthony J. Crowley is. Everyone except a certain bookshop-owning Angel, that is.
And all this because he doodled his initial with his crush's initial in the margins of a report, and had to lie his ass off to keep from being discovered.
(And no, he never tells Aziraphale. How in Hell's name would Crowley ever live it down?)