They had just met, but Harry's initial fun, coupled with Draco's sense of humor had resulted in them becoming tentative friends; they were playing together.
It's not as complicated as it seems.
Harry is getting tired of the response he keeps getting.. "Oh, my god! Is it REALLY Harry Potter!" and the like. He decides he's going to short circuit the process by taking the initiative and doing it himself, to others.
Harry could predict where this conversation was about to go, and he decided in a split second of frustration that enough was enough.
"Good heavens," whispered Harry, "it couldn't be." He let his eyes widen. "Your... name, sir?"
"Draco Malfoy," said Draco Malfoy, looking slightly puzzled.
"It is you! Draco Malfoy. I - I never thought I'd be so honoured, sir." Harry wished he could make tears come out of his eyes. The others usually started crying at around this point.
"Wow," said Draco Malfoy, sounding slightly impressed. The boy gave a sort of wistful sigh. "Your flattery was great, or I thought so, anyway - you'd do well in Slytherin House, too. Usually it's only my father who gets that sort of grovelling. I'm hoping the other Slytherins will suck up to me now I'm at Hogwarts... I guess this is a good sign, then."
Harry coughed. "Actually, sorry, I've got no idea who you are really."
Draco first enjoys the flattery, but when he discovers who Harry is, he teases him by doing the stereotype response (despite being from the 'dark' side of things), then really rubbing it by going fanboy on him.
"What? " said Draco. "Who are you?"
"Harry Potter, pleased to meet you."
"Harry Potter? " gasped Draco. "The Harry -" and the boy cut off abruptly.
There was a brief silence.
Then, with bright enthusiasm, "Harry Potter? The Harry Potter? Gosh, I've always wanted to meet you!"
Draco's attendant emitted a sound like she was strangling but kept on with her work, lifting Draco's arms to carefully remove the chequered robe.
"Shut up," Harry suggested.
"Can I have your autograph? No, wait, I want a picture with you first!"
"I'm just so delighted to meet you!"
"Burst into flames and die."
"But you're Harry Potter, the glorious saviour of the wizarding world! Everyone's hero, Harry Potter! I've always wanted to be just like you when I grow up so I can -"
After this Lucius comes in... and Harry resumes the proven tactic:
"Lucius Malfoy!" gasped Harry Potter. "The Lucius Malfoy?"
One of Malkin's assistants had to turn away and face the wall.
Coolly murderous eyes regarded him. "Harry Potter."
"I am so, so honoured to meet you!"
The dark eyes widened, shocked surprise replacing deadly threat.
It's basically taking the same joke and running with it.. You may notice Malkin's assistants looking away to keep from being seen grinning / laughing.
Several chapters later, they discuss it:
Harry bowed his head, accepting the compliment. "I'm terribly sorry about that," Harry said. "Our first meeting, I mean. I didn't mean to embarrass you in front of Lucius."
Draco waved it off while giving Harry an odd look. "I just wish Father could have come in while you were flattering me -" Draco laughed. "But thank you for what you said to Father. If not for that, I might've had a harder time explaining."
Harry swept a deeper bow. "And thank you for reciprocating with what you said to Professor McGonagall."
"You're welcome. Though one of the assistants must've sworn her closest friend to absolute secrecy, because Father says there're weird rumors going around, like you and I got in a fight or something."
In the end, two brand new (potential) friends were basically playing together; Malfoy confirms this a moment later in chapter 7:
Harry nodded. "I'm glad to hear you're not in trouble."
Draco smirked. "Father has, um, a refined sense of humor, but he does understand making friends. He understands it very well.