Where did the theory of Batman being a mob enforcer come from?

In universe, the reporters in Batman are always scandalous, speculative, both for and against Batman. Specifically, they are prone to:

sen·sa·tion·al·ism. sen?s?SH?nl?iz?m/ noun

1. (especially in journalism) the use of exciting or shocking stories or language at the expense of accuracy, in order to provoke public interest or excitement.

Gotham is a very corrupt city, and it's public is pretty decadent. Mobsters run wild, and the police can barely deal with random crime let alone organized crime. Look at the parade for example. People know that Joker kills, but still show up for free money. Sensationalism is the bread and butter for Gotham media and public really.

In this scene, they have no idea why or who killed the mobsters, the corrupt underground of Gotham, who Batman is known to prey on. So they take a wild guess.

Out of universe, you are right. It's to provoke the Joker, who is always known to get mad when others get credit for his work. Joker tends to kill those people, regardless if it's them or others that claimed that other person did it. He's an attention whore of the worst kind. Very Bruce Vain even. It contrasts Batman, who doesn't care if someone takes credit, as long as it doesn't cause extra harm. He'll care more if that person is opening themselves up to getting hurt (See Nolan verse when random guys use hockey gear to be pretend bat men to their own harm).